Physics | Hydrodynamics » Nicolas Borghini - Elements of Hydrodynamics

Elements of Hydrodynamics Nicolas Borghini Version of September 2, 2015 Nicolas Borghini Universität Bielefeld, Fakultät für Physik Homepage: http://www.physikuni-bielefeldde/~borghini/ Email: borghini at physik.uni-bielefeldde Foreword The following pages were originally not designed to fall under your eyes. They grew up from handwritten notes for myself, listing the important points which I should not forget in the lecture room. As time went by, more and more remarks or developments were added, which is why I started to replace the growingly dirty sheets of paper by an electronic versionthat could then also be easily uploaded on the web page of my lecture, for the benefit(?) of the students. Again, additional results, calculations, comments, paragraphs or even whole chapters accumulated, leading to the temporary outcome which you are reading now: a not necessarily optimal overall outline; at times, unfinished sentences; not fully detailed proofs or calculationsbecause the

missing steps are obvious to me; insufficient discussions of the physics of some resultswhich I hopefully provide in the classroom; not-so-good-looking figures; incomplete bibliography; etc. You may also expect a few solecisms, inconsistent notations, and the usual, unavoidable typos.(∗) Eventually, you will have to cope with the many idiosyncrasies in my writing, as for instance my immoderate use of footnotes, dashes or parentheses, quotation marks, which are not considered as “good practice”. In short: the following chapters may barely be called “lecture notes”; they cannot replace a textbook(†) and the active participation in a course and in the corresponding tutorial/exercise sessions. (∗) (†) Comments and corrections are welcome! . which is one of several good reasons why you should think at least twice before printing a hard copy! Contents Introduction• I • • • • • • • • • • • • Basic notions on continuous media •

• • • • • • • • • • • • • • • • • • • 1 • • • • • • • • • • • • • • • • • • • • 2 I.1 Continuous medium: a model for many-body systems 2 I.11 Basic ideas and concepts 2 I.12 General mathematical framework 4 I.13 Local thermodynamic equilibrium 4 I.2 Lagrangian description 7 I.21 Lagrangian coordinates 8 I.22 Continuity assumptions 8 I.23 Velocity and acceleration of a material point 8 I.3 Eulerian description 9 I.31 I.32 I.33 I.34 Eulerian coordinates. Velocity field 9 Equivalence between the Eulerian and Lagrangian viewpoints 10 Streamlines 10 Material derivative 11 I.4 Mechanical stress 13 I.41 Forces in a continuous medium 13 I.42 Fluids 14 II Kinematics of a continuous medium • • • • • • • • • • • • • • • • • • • • 16 • • • • • • • • • • • • • • • 24

• • • • • • • • • • 25 II.1 Generic motion of a continuous medium 16 II.11 Local distribution of velocities in a continuous medium 17 II.12 Rotation rate tensor and vorticity vector 18 II.13 Strain rate tensor 19 II.2 Classification of fluid flows 22 II.21 Geometrical criteria 22 II.22 Kinematic criteria 22 II.23 Physical criteria 23 Appendix to Chapter II • • • • • • • • • • • • • II.A Deformations in a continuous medium 24 III Fundamental equations of non-relativistic fluid dynamics III.1 Reynolds transport theorem 25 III.11 Closed system, open system 25 III.12 Material derivative of an extensive quantity 26 III.2 Mass and particle number conservation: continuity equation 28 III.21 Integral formulation 28 III.22 Local formulation 29 III.3 Momentum balance: Euler and Navier–Stokes equations 29 III.31 III.32 III.33 III.34 Material derivative of momentum 30 Perfect fluid: Euler equation 30 Newtonian fluid:

Navier–Stokes equation 34 Higher-order dissipative fluid dynamics 38 v III.4 Energy conservation, entropy balance 38 III.41 Energy and entropy conservation in perfect fluids 39 III.42 Energy conservation in Newtonian fluids 40 III.43 Entropy balance in Newtonian fluids 41 IV Non-relativistic flows of perfect fluids • • • • • • • • • • • • • • • • • • • 44 • • • • • • • • • • • • • • • • • 70 • • • • • • • • • • • • • • • • 89 IV.1 Hydrostatics of a perfect fluid 44 IV.11 IV.12 IV.13 IV.14 Incompressible fluid 45 Fluid at thermal equilibrium 45 Isentropic fluid 45 Archimedes’ principle 47 IV.2 Steady inviscid flows 48 IV.21 Bernoulli equation 48 IV.22 Applications of the Bernoulli equation 49 IV.3 Vortex dynamics in perfect fluids 52 IV.31 Circulation of the flow velocity Kelvin’s theorem 52 IV.32 Vorticity

transport equation in perfect fluids 54 IV.4 Potential flows 56 IV.41 Equations of motion in potential flows 56 IV.42 Mathematical results on potential flows 57 IV.43 Two-dimensional potential flows 60 V Waves in non-relativistic perfect fluids • • V.1 Sound waves 70 V.11 Sound waves in a uniform fluid at rest 71 V.12 Sound waves on moving fluids 74 V.13 Riemann problem Rarefaction waves 74 V.2 Shock waves 75 V.21 Formation of a shock wave in a one-dimensional flow 75 V.22 Jump equations at a surface of discontinuity 76 V.3 Gravity waves 79 V.31 Linear sea surface waves 79 V.32 Solitary waves 83 VI Non-relativistic dissipative flows • • • • • VI.1 Statics and steady laminar flows of a Newtonian fluid 89 VI.11 VI.12 VI.13 VI.14 Static Newtonian fluid 89 Plane Couette flow 90 Plane Poiseuille flow 91 Hagen–Poiseuille flow 92 VI.2 Dynamical similarity 94 VI.21 Reynolds number 94 VI.22 Other dimensionless numbers 95 VI.3 Flows at small Reynolds number 96

VI.31 Physical relevance Equations of motion 96 VI.32 Stokes flow past a sphere 97 VI.4 Boundary layer 100 VI.41 Flow in the vicinity of a wall set impulsively in motion 100 VI.42 Modeling of the flow inside the boundary layer 102 VI.5 Vortex dynamics in Newtonian fluids 104 VI.51 Vorticity transport in Newtonian fluids 104 VI.52 Diffusion of a rectilinear vortex 105 VI.6 Absorption of sound waves 106 vi VII Turbulence in non-relativistic fluids • • • • • • • • • • • • • • • • • • • 110 • • • • • • • • • • • • • • 125 • • • • • • • • • • • 133 VII.1 Generalities on turbulence in fluids 110 VII.11 VII.12 VII.13 VII.14 Phenomenology of turbulence 110 Reynolds decomposition of the fluid dynamical fields 112 Dynamics of the mean flow 113 Necessity of a statistical approach 115 VII.2 Model of the turbulent viscosity 116 VII.21 VII.22 VII.23 VII.24

Turbulent viscosity 116 Mixing-length model 117 k -model 118 (k -ε)-model 118 VII.3 Statistical description of turbulence 119 VII.31 Dynamics of the turbulent motion 119 VII.32 Characteristic length scales of turbulence 120 VII.33 The Kolmogorov theory (K41) of isotropic turbulence 122 VIII Convective heat transfer • • • • • • • • • VIII.1 Equations of convective heat transfer 125 VIII.11 Basic equations of heat transfer 125 VIII.12 Boussinesq approximation 127 VIII.2 Rayleigh–Bénard convection 128 VIII.21 Phenomenology of the Rayleigh–Bénard convection 128 VIII.22 Toy model for the Rayleigh–Bénard instability 131 IX Fundamental equations of relativistic fluid dynamics IX.1 Conservation laws 134 IX.11 Particle number conservation 134 IX.12 Energy-momentum conservation 136 IX.2 Four-velocity of a fluid flow Local rest frame 137 IX.3 Perfect relativistic fluid 139 IX.31 Particle four-current and energy-momentum tensor of a perfect fluid 139

IX.32 Entropy in a perfect fluid 141 IX.33 Non-relativistic limit 142 IX.4 Dissipative relativistic fluids 144 IX.41 IX.42 IX.43 IX.44 IX.45 Dissipative currents 144 Local rest frames 147 General equations of motion 148 First order dissipative relativistic fluid dynamics 149 Second order dissipative relativistic fluid dynamics 151 Appendices to Chapter IX • • • • • • • • • • • • • • • • • • • • • • • • • 153 • • • • • • • • • • • • • • 155 IX.A Microscopic formulation of the hydrodynamical fields 153 IX.A1 Particle number 4-current 153 IX.A2 Energy-momentum tensor 154 IX.B Relativistic kinematics 154 IX.C Equations of state for relativistic fluids 154 X Flows of relativistic fluids • • • X.1 Relativistic fluids at rest 155 X.2 One-dimensional relativistic flows 155 X.21 Landau flow 155 X.22 Bjorken flow 155 • • • • • • • vii

Appendices • • • • • • • • • • • • • • • • • • • • • 159 A Basic elements of thermodynamics • • • • • • • • • • • • • • • • • • • 159 B Tensors on a vector space • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 160 • • • • • • • • • • • • • • • • • • • • • • 165 Elements on holomorphic functions of a complex variable • • • • • • • • • 170 • • • • • • • • 173 • B.1 Vectors, one-forms and tensors 160 B.11 B.12 B.13 B.14 Vectors 160 One-forms 160 Tensors 161 Metric tensor 162 B.2 Change of basis 164 C Tensor calculus • • • • • • C.1 Covariant differentiation of tensor fields 165 C.11 C.12 C.13 C.14

Covariant differentiation of vector fields 165 Examples: differentiation in Cartesian and in polar coordinates 167 Covariant differentiation of general tensor fields 168 Gradient, divergence, Laplacian 168 C.2 Beginning of elements of an introduction to differential geometry 169 D D.1 Holomorphic functions 170 D.11 Definitions 170 D.12 Some properties 170 D.2 Multivalued functions 171 D.3 Series expansions 171 D.31 Taylor series 171 D.32 Isolated singularities and Laurent series 171 D.33 Singular points 171 D.4 Conformal maps 172 Bibliography • • • • • • • • • • • • • • • • • • • • • • • viii Introduction General introduction and outline. Notations, conventions, etc. General references (in alphabetical order) • Faber, Fluid dynamics for physicists [1] • Guyon, Hulin, Petit & Mitescu, Physical hydrodynamics [2] • Landau & Lifshitz, Course of theoretical physics. Vol 6: Fluid mechanics [3] =

Landau & Lifschitz, Lehrbuch der theoretischen Physik. Band VI: Hydrodynamik [4] • Sommerfeld, Lectures on theoretical physics. Vol II: Mechanics of deformable bodies [5] = Vorlesungen über theoretische Physik. Band II: Mechanik der deformierbaren Medien [6] C HAPTER I Basic notions on continuous media A system of many microscopic degrees of freedom is often more conveniently described as a material body that fills some region of space continuously, rather than as a collection of discrete points (Sec. I1) This theoretical approach, which is especially suited to represent systems whose internal deformations are relevant, is an instance of physical modeling, originally motivated by the agreement of its predictions with experimental observations. Like every model, that of a continuous medium is valid only in some range of physical conditions, in particular on macroscopic scales. Mathematically, a classical continuous medium at a given instant is described as a volumeor more

generally a manifoldin usual Euclidean space. The infinitesimal elements of this volume constitute the elementary “material points”, which are entirely characterized by their position. To describe the time evolution of the physical system modeled as a continuous medium, two equivalent approaches are available. The first one consists in following the trajectories of the material points as time progresses (Sec. I2) The physical picture of continuousness is then enforced by requesting that the mapping between the position of a given point at some reference initial time and its position at any later instant is continuous. The second point of view, which will mostly be adopted in the remainder of these notes, focuses on the change in the various physical quantities at a fixed position as time elapses (Sec. I3) The reference for the medium evolution between successive instants t and t + dt is the “current” configuration of the material points, i.e at time t, instead of their

positions in the (far) past In that description, the spatial variables are no longer dynamical, but only labels for the position at which some observable is considered. Accordingly, the dynamical quantities in the system are now timedependent fields; the desired continuousness of the medium translates into continuity conditions on those fields. Eventually, the mathematical object that models internal forces in a continuous medium, i.e the influence from neighboring material points on each other, is shortly introduced (Sec. I4) This allows the classification of deformable continuous media into two traditional large classes, and in particular the definition of fluids. I.1 Continuous medium: a model for many-body systems In this Section, we first spell out a few arguments which lead to the introduction of the model of a continuous medium (Sec. I11) The basic ingredients of the mathematical implementation of the model are then presented and a few notions are defined (Sec. I12) Eventually,

the physical assumptions underlying the modeling are reexamined in greater detail, and some more or less obvious limitations of the continuous description are indicated (Sec. I13) I.11 Basic ideas and concepts The actual structure of matter at the microscopic scale is discrete, and involves finite “elementary” entities: electrons, atoms, ions, molecules, . , which in the remainder of these notes will be collectively referred to as “atoms”. Any macroscopic sample of matter contains a large amount of these atoms. For instance, the number density in an ideal gas under normal conditions is about I.1 Continuous medium: a model for many-body systems 3 2.7 × 1025 m−3 , so that one cubic millimeter still contains 27 × 1016 atoms Similarly, even though the number density in the interstellar medium might be as low as about 102 m−3 , any volume relevant for astrophysics, i.e with at least a kilometer-long linear size, involves a large number of atoms. Additionally, these

atoms are in constant chaotic motion, with individual velocities of order 102 –103 m·s−1 for a system at thermal equilibrium at temperature T 300 K. Given a mean free path(i) of order 10−7 m in a gas under normal conditions, each atom undergoes 109 –1010 times per second, i.e its trajectory changes direction constantly when viewed with a macroscopic viewpoint As is seen in Statistical Mechanics, it is in general unnecessary to know the details of the motion of each atom in a macroscopic system: as a matter of fact, there emerge global characteristics, defined as averages, which can be predicted to a high degree of accuracy thanks to the large number of degrees of freedom involved in their determination, despite the chaoticity of the individual atomic behaviors. The macroscopic properties of systems at (global) thermodynamic equilibrium are thus entirely determined by a handful of collective variables, either extensivelike entropy, internal energy, volume, particle number,

total momentum. , or intensiveas eg the respective densities of the various extensive variables, temperature, pressure, chemical potential, average velocity. , where the latter take the same value throughout the system. When thermodynamic equilibrium does not hold globally in a system, there is still the possibility that one may consider that it is valid locally, “at each point” in space. In that situation whose underlying assumptions will be specified in greater detail in Sec. I13the intensive thermodynamic variables characterizing the system macroscopically become fields, which can vary from point to point. More generally, experience shows that it is fruitful to describe a large amount of characteristicsnot only thermodynamic, but also of mechanical nature, like forces and the displacements or deformations they induceof macroscopic bodies as fields. A “continuous medium” is then intuitively a system described by such fields, which should satisfy some (mathematical)

continuity property with respect to the spatial variables that parameterize the representation of the physical system as a geometrical quantity, as will be better specified in Secs. I2 and I3 Assuming the relevance of the model of a medium whose properties are described by continuous fields is often referred to as continuum hypothesis (ii) . The reader should keep in mind that the modeling of a given macroscopic system as a continuous medium does not invalidate the existence of its underlying discrete atomic structure. Specific phenomena will still directly probe the latter, as e.g X-ray scattering experiments That is, the model has limitations to its validity, especially at small wavelengths or high frequencies, where “small” or “high” implies a comparison to some microscopic physical scale characteristic of the system under consideration. Turning the reasoning the other way around, the continuous-medium picture is often referred to as a long-wavelength, low-frequency

approximation to a more microscopic descriptionfrom which it can actually be shown to emerge in the corresponding limits. It is important to realize that the model itself is blind to its own limitations, i.e there is no a priori criterion within the mathematical continuous-medium description that signals the breakdown of the relevance of the picture to actual physics. In practice, there might be hints that the equations of the continuous model are being applied in a regime where they should not, as for instance if they yield negative values for a quantity which should be positive, but such occurrences are not the general rule. Remarks: ∗ The model of a continuous model is not only applicableand appliedto obvious cases like gases, liquids or (deformable) solids, it may also be used to describe the behaviors of large crowds, fish schools, car traffic. provided the number of “elementary” constituents is large and the system is studied on a large enough scale. (i) mittlere freie

Weglänge (ii) Kontinuumshypothese 4 Basic notions on continuous media ∗ Even if the continuous description is valid on “long wavelengths”, it remains obvious that any physical system, viewed on a scale much larger than its spatial extent, is to first approximation best described as pointlike. Consider for instance a molecular cloud of interstellar medium with a 10 parsec radius and about 1010 H2 molecules per cubic meter. For a star forming at its core, it behaves a continuous medium; 1 kpc away, however, the inner degrees of the cloud are most likely already irrelevant and it is best described as a mere point. I.12 General mathematical framework Consider a non-relativistic classical macroscopic physical system Σ, described by Newtonian physics. The positions of its individual atoms, viewed as pointlike, at a given instant twhich is the same for all observersare points in a three-dimensional Euclidean spaceE 3 . In the description as a continuous medium, the system Σ

is represented by a geometrical manifold in E 3 , which for the sake of simplicity will be referred to as a “volume” and denoted by V . The basic constituents of V are its infinitesimal elements d3 V , called material points (iii) or continuous medium particles (iv) which explains a posteriori our designating the discrete constituents of matter as “atoms”, or, in the specific case of the elementary subdivisions of a fluid, fluid particles (v) . As we shall state more explicitly in Sec. I13, these infinitesimal elements are assumed to have the same physical properties as a finite macroscopic piece. Associated with the physical picture attached to the notion of continuousness is the requirement that neighboring material points in the medium remain close to each other throughout the system evolution. We shall see below how this picture is implemented in the mathematical description Remark: The volume V with the topology inherited fromE 3 need not be simply connected. For instance,

one may want to describe the flow of a river around a bridge pier: the latter represents a physical region which water cannot penetrate, which is modeled as a hole throughout the volume V occupied by fluid particles. To characterize the position of a given material point, as well as some of the observables relative to the physical system Σ, one still needs to specify the reference frame in which the system is studied, corresponding to the point of view of a given observer, and to choose a coordinate system in that reference frame. This choice allows one to define vectorslike position vectors, velocities, or forcesand tensors. The basis vectors of the coordinate system will be designated as ~e1 , ~e2 , ~e3 , while the components of a given vector will be denoted with upper (“contravariant”) indices, as e.g ~c = ci ~ei , where the Einstein summation convention over repeated upper and lower indices was used. Once the reference frame and coordinate system are determined, the

macroscopic state of the physical system at time t is mapped onto a corresponding configuration (vi) κt of the medium, consisting of the continuous set of the position vectors ~r = xi ~ei of its constituting material points. Since the volume occupied by the latter may also depends on time, it will also be labeled by t: V t . To be able to formalize the necessary continuity conditions in the following Sections, one also introduces a reference time t0 conveniently taken as the origin of the time axis, t0 = 0and the corresponding reference configuration κ0 of the medium, which occupies a volume V 0 . The generic ~ = X i ~ei . position vector of a material point in this reference configuration will be denoted as R Remark: In so-called “classical” continuous media, as have been introduced here, the material points are entirely characterized by their position vector. In particular, they have no intrinsic angular momentum. (iii) Materielle Punkte (iv) Mediumteilchen (v)

Fluidteilchen (vi) Konfiguration 5 I.1 Continuous medium: a model for many-body systems I.13 Local thermodynamic equilibrium In a more bottom-up approach to the modeling of a system Σ of discrete constituents as a continuous medium, one should first divide Σ (in thought) into small cells of fixedyet not necessarily universalsize fulfilling two conditions: (i) each individual cell can meaningfully be treated as a thermodynamic system, i.e it must be large enough that the relative fluctuation of the usual extensive thermodynamic quantities computed for the content of the cell are negligible; (ii) the thermodynamic properties vary little over the cell scale, i.e cells cannot be too large, so that (approximate) homogeneity is ensured. measured local quantity The rationale behind these two requirements is illustrated by Fig. I1, which represents schematically how the value of a local macroscopic quantity, e.g a density, depends on the resolution of the apparatus with which it is

measured, i.e equivalently on the length scale on which it is defined If the apparatus probes too small a length scale, so that the discrete degrees of freedom become relevant, the measured value strongly fluctuates from one observation to the next one, as hinted at by the displayed envelope of possible results of measurements: this is the issue addressed by condition (i). Simultaneously, a small change in the measurement resolution, even with the apparatus still centered on the same point in the system, can lead to a large variation in the measured value of the observable, corresponding to the erratic behavior of the curve at small scales shown in Fig. I1 This fluctuating pattern decreases with increasing size of the observation scale, since this increase leads to a growth in the number of atoms inside the probed volume, and thus a drop in the size of relative fluctuations. At the other end of the curve, one reaches a regime where the low resolution of the observation leads to

encompassing domains with enough atoms to be rid of fluctuations, yet with inhomogeneous macroscopic properties, in a single probed regionin violation of condition (ii). As a result, the measured value of the density under consideration slowly evolves with the observation scale. In between these two domains of strong statistical fluctuations and slow macroscopic variations lies a regime where the value measured for an observable barely depends on the scale over which it is envelope of the set of possible values macroscopic variation of the local quantity well-defined local value strong variations on “atomic” scale observation scale Figure I.1 – Typical variation of the measured value for a “local” macroscopic observable as a function of the size scale over which it is determined. 6 Basic notions on continuous media determined. This represents the appropriate regime for meaningfully definingand measuringa local density, and more general local quantities. It is

important to note that this intermediate “mesoscopic” interval may not always exist. There are physical systems in which strong macroscopic variations are already present in a range of scales where microscopic fluctuations are still sizable. For such systems, one cannot find scale-independent local variables. That is, the proper definition of local quantities implicitly relies on the existence of a clear separation of scales in the physical system under consideration, which is what will be assumed in the remainder of these notes. Remark: The smallest volume over which meaningful local quantities can be defined is sometimes called representative volume element (vii) (RVE), or representative elementary volume. When conditions (i) and (ii) hold, one may in particular define local thermodynamic variables, corresponding to the values taken in each intermediate-size celllabeled by its position ~rby the usual extensive parameters: internal energy, number of atoms. Since the separation

between cells is immaterial, nothing prevents energy or matter from being transported from a cell to its neighbors, even if the global system is isolated. Accordingly, the local extensive variables in any given cell are actually time-dependent in the general case. In addition, it becomes important to add linear momentumwith respect to some reference frameto the set of local extensive variables characterizing the content of a cell. The size of each cell is physically irrelevant, as long as it satisfies the two key requirements; there is thus no meaningful local variable corresponding to volume. Similarly, the values of the extensive variables in a given cell, which are by definition proportional to the cell size, are as arbitrary as the latter. They are thus conveniently replaced by the respective local densities: internal energy density e(t,~r), number density n (t,~r), linear momentum density ρ(t,~r)~v(t,~r), where ρ denotes the mass density, entropy density s(t,~r). Remark:

Rather than considering the densities of extensive quantities, some authorsin particular Landau & Lifshitz [3, 4]prefer to work with specific quantities, i.e their respective amounts per unit mass, instead of per unit volume. The relation between densities and specific quantities is trivial: denoting by x j resp. x j,m a generic local density resp specific amount for the same physical quantity, one has the identity x j (t,~r) = ρ(t,~r) x j,m (t,~r) (I.1) in every celllabeled by ~rand at every time t. Once the local extensive variables have been meaningfully defined, one can develop the usual formalism of thermodynamics in each cell. In particular, one introduces the conjugate intensive variables, as e.g local temperature T (t,~r) and pressure P (t,~r) The underlying, important hypothesis is the assumption of a local thermodynamic equilibrium According to the latter, the equation(s) of state of the system inside the small cell, expressed with local thermodynamic quantities, is the

same as for a macroscopic system in the actual thermodynamic limit of infinitely large volume and particle number. Consider for instance a non-relativistic classical ideal gas: its (mechanical) equation of state reads PV = N kB T , with N the number of atoms, which occupy a volume V at uniform pressure P and temperature T , while kB is the Boltzmann constant. This is trivially recast as P = n kB T , with n the number density of atoms. The local thermodynamic equilibrium assumption then states that under non-uniform conditions of temperature and pressure, the equation of state in a local cell at position ~r is given by P (t,~r) = n (t,~r)kB T (t,~r) (I.2) at every time t. (vii) Repräsentatives Volumen-Element 7 I.2 Lagrangian description The last step towards the continuous-medium model is to promote ~r, which till now was simply the discrete label attached to a given cell, to be a continuous variable taking its values in R3 or rather, in the volume V t attached to the system

at the corresponding instant t. Accordingly, taking into account the time-dependence of physical quantities, the local variables, in particular the thermodynamic parameters, become fields on R × R3 . The replacement of the fine-resolution description, in which atoms are the relevant degrees of freedom, by the lower-resolution model which assimilates small finite volumes of the former to structureless points is called coarse graining (viii) . This is a quite generic procedure in theoretical physics, whereby the finer degrees of freedom of a more fundamental description are smoothed awaytechnically, this is often done by performing averages or integrals, so that these degrees of freedom are “integrated out”and replaced by novel, effective variables in a theory with a more limited range of applicability, but which is more tractable for “long-range” phenomena. Coming back to condition (ii), we already stated that it implicitly involves the existence of at least one large length

scale L, over which the macroscopic physical properties of the system may vary. This scale can be a characteristic dimension of the system under consideration, as eg the diameter of the tube in which a liquid is flowing. In the case of periodic waves propagating in the continuous medium, L also corresponds to their wavelength. More generally, if G denotes a macroscopic physical quantity, one may consider " # ~ G (t,~r) −1 ∇ , (I.3) L∼ = |G (t,~r)| ~ denotes the (spatial) gradient. where ∇ Condition (i) in particular implies that the typical size of the cells which are later coarse grained should be significantly larger than the mean free path `mfp of atoms, so that thermodynamic equilibrium holds in the local cells. Since on the other hand this same typical size should be significantly smaller than the scale L of macroscopic variations, one deduces the condition Kn ≡ `mfp 1 L (I.4) on the dimensionless Knudsen number Kn.(a) In air under normal conditions P = 105 Pa

and T = 300 K, the mean free path is `mfp ≈ 0.1 µm In the study of phenomena with variations on a characteristic scale L ≈ 10 cm, one finds Kn ≈ 10−6 , so that air can be meaningfully treated as a continuous gas. The opposite regime Kn > 1 is that of a rarefied medium, as for instance of the so-called Knudsen gas, in which the collisions between atoms are negligibleand in particular insufficient to ensure thermal equilibrium as an ideal gas. The flow of such systems is not well described by hydrodynamics, but necessitates alternative descriptions like molecular dynamics, in which the degrees of freedom are explicitly atoms. I.2 Lagrangian description The Lagrangian (b) perspective, which generalizes the approach usually adopted in the description of the motion of a (few) point particle(s), focuses on the trajectories of the material points, where the latter are labeled by their position in the reference configuration. Accordingly, physical quantities ~ and any continuity

condition has are expressed as functions of time t and initial position vectors R, to be formulated with respect to these variables. (viii) (a) Vergröberung M. Knudsen, 1871-1949 (b) J.-L Lagrange, 1736–1813 8 Basic notions on continuous media I.21 Lagrangian coordinates Consider a material point M in a continuous medium. Given a reference frame R, which allows the definition of its position vector at any time t, one can follow its trajectory ~r(t), which, after having chosen a coordinate system, is equivalently represented by the {xi (t)} for i = 1, 2, 3. ~ resp. {X i } denote the position resp coordinates of the material point M at t0 The Let R trajectory obviously depends on this “initial” position, and ~r can thus be viewed as a function of t ~ where the latter refers to the reference configuration κ0 : and R, ~ ~r = ~r(t, R) (I.5a) with the consistency condition ~ = R. ~ ~r(t = t0 , R) (I.5b) In the Lagrangian description, also referred to as material

description or particle description, this point of view is generalized, and the various physical quantities G characterizing a continuous ~ medium are viewed at any time as mathematical functions of the variables t and R: ~ G = G (t, R), (I.6) where the mapping G which as often in physics will be denoted with the same notation as the physical quantity represented by its valueis defined for every t on the initial volume V 0 occupied by the reference configuration κ0 . ~ Together with the time t, the position vector Ror equivalently its coordinates X 1 , X 2 , X 3 in a given systemare called Lagrangian coordinates. I.22 Continuity assumptions ~ is simply the (vector) position An important example of physical quantity, function of t and R, in the reference frame R of material points at time t, i.e ~r or equivalently its coordinates {xi }, as given by relation (I.5a), which thus relates the configurations κ0 and κt ~ maps for every t the initial volume V 0 onto V t . To implement

matheMore precisely, ~r(t, R) matically the physical picture of continuity, it will be assumed that the mapping ~r(t, · ) : V 0 V t is also one-to-one for every ti.e all in all bijective, and that the function ~r and its inverse ~ = R(t,~ ~ r) R (I.7) are continuous with respect to both time and space variables. This requirement in particular ensures that neighboring points remain close to each other as time elapses. It also preserves the connectedness of volumes, (closed) surfaces or curves along the evolution: one may then define material domains, i.e connected sets of material points which are transported together in the evolution of the continuous medium. For the sake of simplicity, it will be assumed that the mapping ~r and its inverse, and more generally every mathematical function G representing a physical quantity, is at least twice continuously differentiable (i.e of class C 2 ) To be able to accommodate for important phenomena that are better modeled with discontinuities,

like shock waves in fluids (Sec. V2) or ruptures in solidsfor instance, in the Earth’s crust, the C 2 -character of functions under consideration may only hold piecewise. I.23 Velocity and acceleration of a material point ~ the function t 7 ~r(t, R) ~ is the trajectory of As mentioned above, for a fixed reference position R ~ the material point which passes through R at the reference time t0 . As a consequence, the velocity in the reference frame R of this same material point at time t is simply ~ ~ = ∂~r(t, R) . ~v (t, R) (I.8) ∂t 9 I.3 Eulerian description ~ is independent of t, one could actually also write ~v (t, R) ~ = d~r(t, R)/dt. ~ Since the variable R In turn, the acceleration of the material point in R is given at time t by ~ ~ = ∂~v (t, R) . ~a(t, R) ∂t (I.9) Remark: The trajectory (or pathline (ix) ) of a material point can be visualized, by tagging the point ~ for instance with a fluorescent or radioactive marker, and then imaging at time t0 at its

position R, the positions at later times t > t0 . If on the other hand one regularlysay for every instant t0 ≤ t0 ≤ tinjects some marker at a fixed geometrical point P , the resulting tagged curve at time t is the locus of the geometrical points occupied by medium particles which passed through P in the past. This locus is referred to as streakline.(x) Denoting by ~rP the position vector of point P , the streakline is the set of geometrical points with position vectors
~ 0 ,~rP ) for t0 ≤ t0 ≤ t. ~r = ~r t, R(t (I.10) I.3 Eulerian description The Lagrangian approach introduced in the previous Section is actually not commonly used in fluid dynamics, at least not in its original form, except for specific problems. One reason is that physical quantities at a given time are expressed in terms of a reference configuration in the (far) past: a small uncertainty on this initial condition may actually yield after a finite duration a large uncertainty on the present state of the

system, which is problematic. On the other hand, this line of argument explains why the Lagrangian point of view is adopted to investigate chaos in many-body systems! The more usual description is the so-called Eulerian (c) perspective, in which the evolution between instants t and t + dt takes the system configuration at time t as a reference. I.31 Eulerian coordinates Velocity field In contrast to the “material” Lagrangian point of view, which identifies the medium particles in a reference configuration and follows them in their evolution, in the Eulerian description the emphasis is placed on the geometrical points. Thus, the Eulerian coordinates are time t and a spatial vector ~r, where the latter does not label the position of a material point, but rather that of a geometrical point. Accordingly, the physical quantities in the Eulerian specification are described by fields on space-time. Thus, the fundamental field that entirely determines the motion of a continuous medium in

a given reference frame R is the velocity field ~vt (t,~r). The latter is defined such that it gives the value of the Lagrangian velocity ~v [cf. Eq (I8)] of a material point passing through ~r at time t: ~v =~vt (t,~r) ∀t, ∀~r ∈ V t . (I.11) More generally, the value taken at given time and position by a physical quantity G , whether attached to a material point or not, is expressed as a mathematical function Gt of the same Eulerian variables: G = Gt (t,~r) ∀t, ∀~r ∈ V t . (I.12) ~ 7 G (t, R) ~ in the Lagrangian approach and (t,~r) 7 G (t,~r) in the Note that the mappings (t, R) t Eulerian description are in general different. For instance, the domains in R3 over which their spatial (ix) (c) Bahnlinie (x) Streichlinie L. Euler, 1707–1783 10 Basic notions on continuous media variables take their values differ: constant (V 0 ) in the Lagrangian specification, time-dependent (V t ) in the case of the Eulerian quantities. Accordingly the latter will be denoted

with a subscript t I.32 Equivalence between the Eulerian and Lagrangian viewpoints Despite the different choices of variables, the Lagrangian and Eulerian descriptions are fully equivalent. Accordingly, the prevalence in practice of the one over the other is more a technical issue than a conceptual one. Thus, it is rather clear that the knowledge of the Lagrangian specification can be used to obtain ~ r) between present and reference the Eulerian formulation at once, using the mapping ~r 7 R(t,~ positions of a material point. Thus, the Eulerian velocity field can be expressed as
~ r) . ~vt (t,~r) = ~v t, R(t,~ (I.13a) This identity in particular shows that ~vt automatically inherits the smoothness properties of ~v : if ~ 7 ~r(t, R) ~ and its inverse are piecewise C 2 (cf. Sec I22), then ~vt is (at least) the mapping (t, R) 1 piecewise C in both its variables. For a generic physical quantity, the transition from the Lagrangian to the Eulerian point of view similarly reads
~ r) . Gt

(t,~r) = G t, R(t,~ (I.13b) Reciprocally, given a (well-enough behaved) Eulerian velocity field ~vt on a continuous medium, one can uniquely obtain the Lagrangian description of the medium motion by solving the initial value problem  ~
  ∂~r(t, R) =~v t,~r(t, R) ~ t ∂t (I.14a)   ~ ~ ~r(t0 , R) = R, where the second line represents the initial condition. That is, one actually reconstructs the pathline of every material point of the continuous medium. Introducing differential notations, the above system can also be rewritten as ~ = R. ~ d~r =~vt (t,~r) dt with ~r(t0 , R) (I.14b) ~ are known, one obtains the Lagrangian function G (t, R) ~ for a given Once the pathlines ~r(t, R) physical quantity G by writing
~ = G t,~r(t, R) ~ . G (t, R) (I.14c) t Since both Lagrangian and Eulerian descriptions are equivalent, we shall from now on drop the subscript t on the mathematical functions representing physical quantities in the Eulerian point of view. I.33 Streamlines At

a given time t, the streamlines (xi) of the motion are defined as the field lines of ~vt . That is, these are curves whose tangent is everywhere parallel to the instantaneous velocity field at the same geometrical point. Let ~x(λ) denote a streamline, parameterized by λ. The definition can be formulated as
d~x(λ) = α(λ)~v t, ~x(λ) dλ (I.15a) with α(λ) a scalar function. Equivalently, denoting by d~x(λ) a differential line element tangent to (xi) Stromlinien 11 I.3 Eulerian description the streamline, one has the condition
d~x ×~v t, ~x(λ) = ~0. (I.15b) Eventually, introducing a Cartesian system of coordinates, the equation for a streamline is conveniently rewritten as dx2 (λ) dx3 (λ) dx1 (λ)
=
=
(I.15c) v1 t, ~x(λ) v2 t, ~x(λ) v3 t, ~x(λ) in a point where none of the component vi of the velocity field vanishesif one of the vi is zero, then so is the corresponding dxi , thanks to Eq. (I15b) Remark: Since the velocity field ~v depends on the choice of

reference frame, this is also the case of its streamlines at a given instant. Consider now a closed geometrical curve in the volume V t occupied by the continuous medium at time t. The streamlines tangent to this curve form in the generic case a tube-like surface, called stream tube.(xii) Let us introduce two further definitions related to properties of the velocity field: • If ~v(t,~r) has at some t the same value in every geometrical point ~r of a (connected) domain D ⊂ V t , then the velocity field is said to be uniform across D . In that case, the streamlines are parallel to each other over D . • If~v(t,~r) only depends on the position, not on time, then the velocity field and the corresponding motion of the continuous medium are said to be steady or equivalently stationary. In that case, the streamlines coincide with the pathlines and the streaklines. Indeed, one checks that Eq. (I14b) for the pathlines, in which the velocity becomes timeindependent, can then be recast (in

a point where all vi are non-zero) as dx2 dx3 dx1 = = , v1 (t,~r) v2 (t,~r) v3 (t,~r) where the variable t plays no role: this is exactly the system (I.15c) defining the streamlines at time t. The equivalence between pathlines and streaklines is also trivial  I.34 Material derivative Consider a material point M in a continuous medium, described in a reference frame R. Let ~r resp. ~r + d~r denote its position vectors at successive instants t resp t + dt The velocity of M at time t resp. t + dt is by definition equal to the value of the velocity field at that time and at the respective position, namely ~v(t,~r) resp. ~v(t + dt,~r + d~r) For small enough dt, the displacement d~r of the material point between t and t+dt is simply related to its velocity at time t by d~r =~v(t,~r) dt. Let d~v ≡ ~v(t + dt,~r + d~r) −~v(t,~r) denote the change in the material point velocity. Assuming that ~v(t,~r) is differentiable (cf. Sec I32) and introducing for simplicity a system of Cartesian

coordinates, a Taylor expansion to lowest order yields d~v ∂~v(t,~r) ∂~v(t,~r) 1 ∂~v(t,~r) 2 ∂~v(t,~r) 3 dt + dx + dx + dx , ∂t ∂x1 ∂x2 ∂x3 up to terms of higher order in dt or d~r. Introducing the differential operator ~ = dx1 ∂ + dx2 ∂ + dx3 ∂ , d~r · ∇ ∂x1 ∂x2 ∂x3 (xii) Stromröhre 12 Basic notions on continuous media this can be recast in the more compact form d~v
∂~v(t,~r) ~ ~v(t,~r). dt + d~r · ∇ ∂t (I.16) In the second term on the right-hand side, d~r can be replaced by ~v(t,~r) dt. On the other hand, the change in velocity of the material point between t and t + dt is simply the product of its acceleration ~a(t) at time t by the size dt of the time interval, at least to lowest order in dt. Dividing both sides of Eq. (I16) by dt and taking the limit dt 0, in particular in the ratio d~v/dt, yield ~a(t) =  ∂~v(t,~r)  ~ ~v(t,~r). + ~v(t,~r) · ∇ ∂t (I.17) That is, the acceleration of the material point consists of

two terms: ∂~v , which follows from the non-stationarity of the velocity field; ∂t
~ ~v, due to the non-uniformity of the motion. • the convective acceleration ~v · ∇ • the local acceleration More generally, one finds by repeating the same derivation that the time derivative of a physical quantity G attached to a material point or domain, yet expressed in terms of Eulerian fields, is the ~ G ] part, irrespective of the tensorial nature of G . sum of a local (∂ G /∂t) and a convective [(~v · ∇) Accordingly, one introduces the operator ∂ D ~ ≡ +~v(t,~r) · ∇ Dt ∂t (I.18) called material derivative (xiii) or (between others) substantial derivative,(xiv) derivative following the motion, hydrodynamic derivative. Relation (I17) can thus be recast as ~a(t) = D~v(t,~r) . Dt (I.19) Remarks: ∗ Equation (I.17) shows that even in the case of a steady motion, the acceleration of a material point may be non-vanishing, thanks to the convective part. ∗ The

material derivative (I.18) is also often denoted (and referred to) as total derivative d/dt ∗ One also finds in the literature the denomination convective derivative.(xv) To the eyes and ears of the author of these lines, that name has the drawback that it does not naturally evoke the local part, but only. the convective one, which comes from the fact that matter is being transported, “conveyed”, with a non-vanishing velocity field ~v(t,~r). ∗ The two terms in Eq. (I18) actually “merge” together when considering the motion of a material point in Galilean space-time R × R3 . As a matter of fact, one easily shows that D/Dt is the (Lie) derivative along the world-line of the material point The world-line element corresponding to the motion between t and t+dt goes from (t, x1 , x2 , x3 ) to (t+dt, x1 +v1 dt, x2 +v2 dt, x3 +v3 dt). The tangent vector to this world-line thus has components (1, v1 , v2 , v3 ), i.e the derivative along the direction of this vector is ∂t + v1

∂1 + v2 ∂2 + v3 ∂3 , with the usual shorthand notation ∂i ≡ ∂/∂xi .  (xiii) Materielle Ableitung (xiv) Substantielle Ableitung (xv) Konvektive Ableitung 13 I.4 Mechanical stress I.4 Mechanical stress I.41 Forces in a continuous medium Consider a closed material domain V inside the volume V t occupied by a continuous medium, and let S denote the (geometric) surface enclosing V. One distinguishes between two classes of forces acting on this domain: • Volume or body forces,(xvi) which act in each point of the bulk volume of V. Examples are weight, long-range electromagnetic forces or, in non-inertial reference frames, fictitious forces (Coriolis, centrifugal). For such forces, which tend to be proportional to the volume they act on, it will later be more convenient to introduce the corresponding volumic force density. • Surface or contact forces,(xvii) which act on the surface S, like friction, which we now discuss in further detail. d2 S ~en d2 F~s V S

Figure I.2 Consider an infinitesimally small geometrical surface element d2 S at point P . Let d2 F~s denote the surface force through d2 S. That is, d2 F~s is the contact force, due to the medium exterior to V, that a “test” material surface coinciding with d2 S would experience. The vector d2 F~s T~s ≡ 2 , d S (I.20) representing the surface density of contact forces, is called (mechanical) stress vector (xviii) on d2 S. The corresponding unit in the SI system is the Pascal, with 1 Pa = 1 N·m−2 . Purely geometrically, the stress vector T~s on a given surface element d2 S at a given point can be decomposed into two components, namely • a vector orthogonal to plane tangent in P to d2 S, the so-called normal stress (xix) ; when it is directed towards the interior resp. exterior of the medium domain being acted on, it also referred to as compression (xx) resp. tension (xxi) ; • a vector in the tangent plane in P , called shear stress (xxii) and often denoted as ~τ .

Despite the short notation adopted in Eq. (I20), the stress vector depends not only on the position of the geometrical point P where the infinitesimal surface element d2 S lies, but also on the (xvi) (xx) Volumenkräfte Druckspannung (xvii) (xxi) (xviii) Oberflächenkräfte Mechanischer Spannungsvektor (xxii) Zugsspannung Scher-, Tangential- oder Schubspannung (xix) Normalspannung 14 Basic notions on continuous media orientation of the surface. Let ~en denote the normal unit vector to the surface element, directed towards the exterior of the volume V (cf. Fig I2), and let ~r denote the position vector of P in a given reference frame. The relation between ~en and the stress vector T~s on d2 S is then linear: T~s = σ(~r) · ~en , (I.21a) with σ (~r) a symmetric tensor of rank 2, the so-called (Cauchy(d) ) stress tensor .(xxiii) In a given coordinate system, relation (I.21a) yields Tsi = 3 X σ ij ejn (I.21b) j=1 with Tsi resp. ejn the coordinates of the vectors T~s

resp ~en , and σ ij the tensor. 1 1
-components of the stress While valid in the case of a three-dimensional position space, equation (I.21a) should actually be better formulated to become valid in arbitrary dimension. Thus, the unit-length “normal vector” to a surface element at point P is rather a 1-form acting on the vectors of the tangent space to the surface at P . As such, it should be represented as the transposed of a vector [(~en )T ], which multiplies the stress tensor from the left: T~s = (~en )T · σ (~r). (I.21c)
This shows that the Cauchy stress tensor is a 20 -tensor (a “bivector”), which maps 1-forms onto vectors. In terms of coordinates, this gives, using Einstein’s summation convention Tsj = en,i σ ij , (I.21d) which thanks to the symmetry of σ is equivalent to the relation given above. Remark: The symmetry property of the Cauchy stress tensor is intimately linked to the assumption that the material points constituting the continuous medium

have no intrinsic angular momentum. I.42 Fluids With the help of the notion of mechanical stress, we may now introduce the definition of a fluid , which is the class of continuous media whose motion is described by hydrodynamics: A fluid is a continuous medium that deforms itself as long as it is submitted to shear stresses. (I.22) Turning this definition around, one sees that in a fluid at restor, to be more accurate, studied in a reference frame with respect to which it is at restthe mechanical stresses are necessarily normal. That is, the stress tensor is in each point diagonal More precisely, for a locally isotropic fluidwhich means that the material points are isotropic,
2 which is the case throughout these notesthe stress 0 -tensor is everywhere proportional to the inverse metric tensor: σ(t,~r) = −P (t,~r) g−1 (t,~r) with P (t,~r) the hydrostatic pressure at position ~r at time t. Going back to relation (I.21b), the stress vector will be
parallel to the “unit normal

vector” in any coordinate system if the square matrix of the 11 -components σ ij is proportional to the (xxiii) (d) (Cauchy’scher ) Spannungstensor A.L Cauchy, 1789–1857 (I.23) I.4 Mechanical stress 15 identity matrix, i.e σ ij ∝ δ ij , where we have introduced the Kronecker symbol To obtain the
2 -components σ ik , one has to multiply σ i by the component g jk of the inverse metric tensor, 0 j summing over k, which precisely gives Eq. (I23) Remarks: ∗ Definition (I.22), as well as the two remarks hereafter, rely on an intuitive picture of “deformations” in a continuous medium To support this picture with some mathematical background, we shall introduce in Sec. IIA an appropriate strain tensor, which quantifies these deformations, at least as long as they remain small. ∗ A deformable solid will also deform itself when submitted to shear stress! However, for a given fixed amount of tangential stress, the solid will after some time reach a new, deformed

equilibrium positionotherwise, it is not a solid, but a fluid. ∗ The previous remark is actually a simplification, valid on the typical time scale of human beings. Thus, materials which in our everyday experience are solidsas for instance those forming the mantle of the Earth will behave on a longer time scale as fluidsin the previous example, on geological time scales. Whether a given substance behaves as a fluid or a deformable solid is sometimes characterized by the dimensionless Deborah number [7], which compares the typical time scale for the response of the substance to a mechanical stress and the observation time. ∗ Even nicer, the fluid vs. deformable solid behavior may actually depend on the intensity of the applied shear stress: ketchup! Bibliography for Chapter I • National Committee for Fluid Mechanics films & film notes on Eulerian Lagrangian description and on Flow visualization;(1) • Faber [1] Chapter 1.1–13; • Feynman [8, 9] Chapter 31–6; • Guyon et

al. [2] Chapter 11; • Sedov [10] Chapters 1 & 2.1–22; • Sommerfeld [5, 6] beginning of Chapter II.5 (1) The visualization techniques have probably evolved since the 1960s, yet pathlines, streaklines or streamlines are still defined in the same way. C HAPTER II Kinematics of a continuous medium The goal of fluid dynamics is to investigate the motion of fluids under consideration of the forces at play, as well as to study the mechanical stresses exerted by moving fluids on bodies with which they are in contact. The description of the motion itself, irrespective of the forces, is the object of kinematics. The possibilities for the motion of a deformable continuous medium, in particular of a fluid, are richer than for a mere point particle or a rigid body: besides translations and global rotations, a deformable medium may also rotate locally and undergo. deformations! The latter term actually encompasses two different yet non-exclusive possibilities, namely either a

change of shape or a variation of the volume. All these various types of motion are encoded in the local properties of the velocity field at each instant (Sec. II1) Generic fluid motions are then classified according to several criteria, especially taking into account kinematics (Sec. II2) For the sake of reference, the characterization of deformations themselves, complementing that of their rate of change, is briefly presented in Sec. IIA That formalism is not needed within fluid dynamics, but rather for the study of deformable solids, like elastic ones. II.1 Generic motion of a continuous medium Let ~v denote the velocity field in a continuous medium, with respect to some reference frame R. To illustrate (some of) the possible motions that occur in a deformable body, Fig. II1 shows the positions at successive instants t and t+δt of a small “material vector” δ~`(t), that is, a continuous set x2 ~v t,~r + δ~`(t)
δ~`(t + δt) δ~`(t) ~v(t,~r) ~r x1 x3 Figure II.1 –

Positions of a material line element δ~` at successive times t and t + δt 17 II.1 Generic motion of a continuous medium of material points distributed along the (straight) line element stretching between two neighboring geometrical points. positions Let ~r and ~r + δ~`(t) denote the geometrical endpoints of this material vector at time t. ~ 7 ~r(t,~r) and its inverse ~r 7 R(t,~ ~ r), the material Thanks to the continuity of the mappings R vector defined at instant t remains a connected set of material points as time evolves, in particular at t + δt. Assuming that both the initial length |δ~`(t)| as well as δt are small enough, the evolved set at t + δt remains approximately along a straight line, and constitutes a new material vector, denoted by δ~`(t+dt). The position vectors of these endpoints simply follow from the initial positions of the corresponding material points: ~r resp. ~r + δ~`(t), to which should be added the respective
displacement vectors between t and

t+δt, namely the product by δt of the initial velocity ~v t,~r
resp. ~v t,~r + δ~`(t) That is, one finds


  (II.1) δ~`(t + δt) = δ~`(t) + ~v t,~r + δ~`(t) −~v t,~r δt + O δt2 . Figure II.1 already suggests that the motion of the material vector consists not only of a translation, but also of a rotation, as well as an “expansion”the change in length of the vector. II.11 Local distribution of velocities in a continuous medium Considering first a fixed time t, let ~v(t,~r) resp. ~v(t,~r) + δ~v be the velocity at the geometric point situated at position ~r resp. at ~r + δ~r in R Introducing for simplicity a system of Cartesian coordinates in R, the Taylor expansion of the i-th component of the velocity fieldwhich is at least piecewise C 1 in its variables, see Sec. I32 gives to first order 3 X ∂ vi (t,~r) j i δv δx . (II.2a) ∂xj j=1 ~ vv(t,~r) whose components in the coordinate system used here are the Introducing the 11 -tensor ∇~ i partial derivatives

∂ v (t,~r)/∂xj , the above relation can be recast in the coordinate-independent form ~ vv(t,~r) · δ~r. δ~v ∇~ (II.2b)
~ vv(t,~r) at time t and position ~r can be Like every rank 2 tensor, the velocity gradient tensor ∇~ decomposed into the sum of the symmetric and an antisymmetric part: ~ vv(t,~r) = D (t,~r) + R (t,~r), ∇~ (II.3a) where one conventionally writes   T  T  1 ~ 1 ~ D(t,~r) ≡ ∇~vv(t,~r) + ~∇~vv(t,~r) , ∇~vv(t,~r) − ~∇~vv(t,~r) R(t,~r) ≡ (II.3b) 2 2   ~ vv(t,~r) T the transposed tensor to ∇~ ~ vv(t,~r). These definitions are to be understood as follows: with ∇~ Using the same Cartesian
coordinate system as above, the components of the two tensors D , R , viewed for simplicity as 02 -tensors, respectively read     1 ∂ vi (t,~r) ∂ vj (t,~r) 1 ∂ vi (t,~r) ∂ vj (t,~r) , R ij (t,~r) = . (II.3c) + − D ij (t,~r) = 2 ∂xj ∂xi 2 ∂xj ∂xi Note that here we have silently used the fact that for Cartesian coordinates, the

positionsubscript or exponentof the index does not change the value of the component, i.e numerically vi = vi for every i ∈ {1, 2, 3}. Relations (II.3c) clearly represent the desired symmetric and antisymmetric parts However, one sees that the definitions would not appear to fulfill their task if both indices were not both 18 Kinematics of a continuous medium either up or down, as e.g D ij (t,~r)   1 ∂ vi (t,~r) ∂ vj (t,~r) + = 2 ∂xj ∂xi in which the symmetry is no longer obvious. The trick is to rewrite the previous identity as     1 ik ∂ vk (t,~r) ∂ vl (t,~r) 1 ik l ∂ vk (t,~r) ∂ vl (t,~r) i l + + = g (t,~r)g j (t,~r) , D j (t,~r) = δ δ j 2 ∂xl ∂xk 2 ∂xl ∂xk where we have used the fact that the metric tensor of Cartesian coordinates coincides with the Kronecker symbol. To fully generalize to curvilinear coordinates, the partial derivatives in the rightmost term should be replaced by the covariant derivatives discussed in Appendix C.1, leading

eventually to   1 ik dvk (t,~r) dvl (t,~r) i l D j (t,~r) = g (t,~r)g j (t,~r) (II.4a) + 2 dxl dxk   dvk (t,~r) dvl (t,~r) 1 ik i l − (II.4b) R j (t,~r) = g (t,~r)g j (t,~r) 2 dxl dxk With these new forms, which are valid in any coordinate system, the raising or lowering of indices does not affect the visual symmetric or antisymmetric aspect of the tensor. Using the tensors D and R we just introduced, whose physical meaning will be discussed at length in Secs. II12–II13, relation (II2b) can be recast as


~v t,~r + δ~r =~v t,~r + D (t,~r) · δ~r + R (t,~r) · δ~r + O |δ~r|2 (II.5) where everything is at constant time. Under consideration of relation (II.5) with δ~r = δ~`(t), Eq (II1) for the time evolution of the material line element becomes  
δ~`(t + δt) = δ~`(t) + D (t,~r) · δ~`(t) + R (t,~r) · δ~`(t) δt + O δt2 . )· Subtracting δ~`(t) from both sides, dividing by δt and taking the limit δt 0, one finds for the rate of change of the material

vector, which is here denoted by a dot: δ~` (t) = D (t,~r) · δ~`(t) + R (t,~r) · δ~`(t) (II.6) In the following two subsections, we shall investigate the physical content of each of the tensors R (t,~r) and D (t,~r). II.12 Rotation rate tensor and vorticity vector The tensor R (t,~r) defined by Eq. (II3b) is called, for reasons that will become clearer below, rotation rate tensor .(xxiv) By construction, this tensor is antisymmetric. Accordingly, one can naturally associate with it ~ r), such that for any vector V ~ a dual (pseudo)-vector Ω(t,~

~ =Ω ~ t,~r × V ~ ∀V ~ ∈ R3 . R t,~r · V ~ r) are related to those of the rotation rate tensor In Cartesian coordinates, the components of Ω(t,~ by 3 1 X ijk Ωi (t,~r) ≡ −  R jk (t,~r) (II.7a) 2 j,k=1 (xxiv) Wirbeltensor 19 II.1 Generic motion of a continuous medium
with ijk the totally antisymmetric Levi-Civita symbol. Using the antisymmetry of R t,~r , this equivalently reads R23 (t,~r), Ω1 (t,~r)

≡ −R R31 (t,~r), Ω2 (t,~r) ≡ −R R12 (t,~r). Ω3 (t,~r) ≡ −R (II.7b) Comparing with Eq. (II3c), one finds ~ r) = 1 ∇ ~ ×~v(t,~r). Ω(t,~ 2 (II.8) )· ~ r), assuming that D (t,~r) Let us now rewrite relation (II.6) with the help of the vector Ω(t,~ vanishes so as to isolate the effect of the remaining term. Under this assumption, the rate of change of the material vector between two neighboring points reads ~ r) × δ~`(t). δ~` (t) = R (t,~r) · δ ~ `(t) = Ω(t,~ (II.9) The term on the right hand side is then exactly the rate of rotation of a vector ~`(t) in the motion ~ r). Accordingly, the pseudovector Ω(t,~ ~ r) is of a rigid body with instantaneous angular velocity Ω(t,~ (xxv) referred to as local angular velocity. This a posteriori justifies the denomination rotation rate tensor for the antisymmetric tensor R (t,~r). Remarks: ~ r), one also defines the vorticity vector (xxvi) as the curl of ∗ Besides the local angular velocity Ω(t,~ the

velocity field ~ r) = ∇ ~ ×~v(t,~r). ω ~ (t,~r) ≡ 2 Ω(t,~ (II.10) In fluid mechanics, the vorticity is actually more used than the local angular velocity. ~ r) or equivalently the vorticity vector ω ∗ The local angular velocity Ω(t,~ ~ (t,~r) define, at fixed t, ~ ~ divergence-free (pseudo)vector fields, since obviously ∇ · (∇ ×~v) = 0. The corresponding field lines are called vorticity lines (xxvii) and are given by [cf. Eq (I15)] d~x × ω ~ (t,~r) = ~0 (II.11a) or equivalently, in a point where none of the components of the vorticity vector vanishes, dx1 1 ω (t,~r) = dx2 2 ω (t,~r) = dx3 . 3 ω (t,~r) (II.11b) II.13 Strain rate tensor According to the previous subsection, the local rotational motion of a material vector is governed by the (local and instantaneous) rotation rate tensor R (t,~r). In turn, the translational motion is simply the displacementwhich must be described in an affine space, not a vector oneof one of the endpoints of δ~` by an

amount given by the product of velocity and length of time interval. That is, both components of the motion of a rigid body are already accounted for without invoking the symmetric tensor D(t,~r). In other words, the tensor D (t,~r) characterizes the local deviation between the velocity fields in ~ r). a deformable body, in particular a fluid, and in a rigid body rotating with angular velocity Ω(t,~ (xxviii) Accordingly, it is called strain rate tensor or deformation rate tensor . As we shall now see, the diagonal and off-diagonal components of D (t,~r) actually describe the rates of change of different kinds of deformation. For simplicity, we assume throughout this ~ r) = ~0. subsection that Ω(t,~ (xxv) (xxvi) Wirbelvektor Wirbligkeit Deformationsgeschwindigkeitstensor (xxvii) Wirbellinien (xxviii) Verzerrungsgeschwindigkeitstensor, 20 Kinematics of a continuous medium II.13 a Diagonal components :::::::::::::::::::::::::::::: We first assume that all off-diagonal terms

in the strain rate tensor vanish: D ij (t,~r) = 0 for i 6= j, so as to see the meaning of the diagonal components. Going back to Eq. (II1), let us simply project it along one of the axes of the coordinate system, say along direction i. Denoting the i-th component of δ~` as δ`i , one thus finds 


δ`i (t + δt) = δ`i (t) + vi t,~r + δ~`(t) − vi t,~r δt + O δt2 . Taylor-expanding the term between square brackets to first order then yields δ`i (t + δt) − δ`i (t) 3 X ∂ vi (t,~r) j=1 ∂xj δ`j (t) δt, ~ r)or equivalently up to terms of higher order in |δ~`(t)| or δt. Since we have assumed that both Ω(t,~ the components R ij (t,~r) of the rotation rate tensorand the off-diagonal D ij (t,~r) with i 6= j vanish, one checks that the partial derivative ∂ vi (t,~r)/∂xj vanishes for i 6= j. That is, there is only the term j = i in the sum, so that the equation simplifies to ∂vi (t,~r) i δ` (t) δt = D ii (t,~r) δ`i (t) δt. ∂xi Thus, the relative

elongation of the i-th componentremember that there is no local rotation, so that the change in δ`i is entirely due to a variation of the length of the material vectorin δt is given by δ`i (t + δt) − δ`i (t) (II.12) = D ii (t,~r) δt. δ`i (t) δ`i (t + δt) − δ`i (t) This means that the diagonal component D ii (t,~r) represents the local rate of linear elongation in direction i. Volume expansion rate Instead of considering a one-dimensional material vector, one can study the evolution of a small “material parallelepiped” of the continuous medium, situated at t at position ~r with instantaneous side lengths δL1 (t), δL2 (t), δL3 (t)for simplicity, the coordinate axes are taken along the sides of the parallelepiped. Accordingly, its volume at time t is simply δV(t) = δL1 (t) δL2 (t) δL3 (t) Taking into account Eq. (II12) for the relative elongation of each side length, one finds that the relative change in volume between t and t + δt is δV(t+δt) − δV(t)

δL1 (t+δt) − δL1 (t) δL2 (t+δt) − δL2 (t) δL3 (t+δt) − δL3 (t) = + + δV(t) δL1 (t) δL2 (t) δL3 (t)   1 = D 1 (t,~r) + D 22 (t,~r) + D 33 (t,~r) δt. In the second line, one recognizes the trace of the tensor D (t,~r), which going back to the definition of the latter is equal to the divergence of the velocity fluid: ∂ v1 (t,~r) ∂ v2 (t,~r) ∂ v3 (t,~r) ~ + + = ∇ · ~v(t,~r). ∂x1 ∂x2 ∂x3 That is, this divergence represents the local and instantaneous volume expansion rate of the continuous medium. Accordingly, the flow of a fluid is referred to as incompressible in some region when the velocity field in that region is divergence free: D 11 (t,~r) + D 22 (t,~r) + D 33 (t,~r) = ~ · ~v(t,~r) = 0. incompressible flow ⇔ ∇ (II.13) II.13 b Off-diagonal components ::::::::::::::::::::::::::::::::: Let us now assume that D 12 (t,~r), and thereby automatically D 21 (t,~r), is the only non-vanishing component of the strain rate tensor. To see the influence

of that component, we need to consider 21 II.1 Generic motion of a continuous medium x2 x2 δ`2 x1 δ`1 v2 δt 1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 δα2 1111111 0000000 1111111 0000000 1111111 (v2 +δv2 )δt δα1 0000000 1111111 0000000 1111111 (v1 +δv1111111 0000000 1 )δt x1 v1 δt Figure II.2 – Evolution of a material rectangle caught in the motion of a continuous medium between times t (left) and t + δt (right). the time evolution of a different object than a material vector, since anything that can affect the lattertranslation, rotation, dilatationhas already been described above. Accordingly, we now look at the change between successive instants t and t + δt of an elementary “material rectangle”, as pictured in Fig. II2 We denote by ~v resp ~v + δ~v the velocity at time t at the lower left resp. upper right corner of the rectangle Taylor expansions give

for the components of the shift δ~v ∂ v2 (t,~r) 1 ∂ v1 (t,~r) 2 δ` , δv2 = d` . δv1 = 2 ∂x ∂x1 Figure II.2 shows that what was at time t a right angle becomes an angle π/2−δα at t+dt, where δα = δα1 + δα2 . In the limit of small δt, both δα1 and δα2 will be small and thus approximately equal to their respective tangents. Using the fact that the parallelogram still has the same area since the diagonal components of D vanishthe projection of any side of the deformed rectangle at time t + δt on its original direction at time t keeps approximately the same length, up to corrections of order δt. One thus finds for the oriented angles δα1 δv2 δt δ`1 and δα2 − δv1 δt . δ`2 With the Taylor expansions given above, this leads to δα1 ∂ v2 (t,~r) δt, ∂x1 δα2 − ∂ v1 (t,~r) δt. ∂x2 Gathering all pieces, one finds δα ∂ v2 (t,~r) ∂ v1 (t,~r) − = 2 D21 (t,~r). δt ∂x1 ∂x2 (II.14) Going to the limit δt 0, one sees

that the off-diagonal component D 21 (t,~r) represents half the local velocity of the “angular deformation”the shear around direction x3 . Remark: To separate the two physical effects present in the strain rate tensor, it is sometimes written as the
sum of a diagonal rate-of-expansion tensor proportional to the identity 1 which is in fact the 11 -form of the metric tensor g of Cartesian coordinatesand a traceless rate-of-shear tensor S :  1 ~ · ~v(t,~r) 1 + S (t,~r) (II.15a) D (t,~r) = ∇ 3 with       1 ~ ~ vv(t,~r) T − 2 ∇ ~ · ~v(t,~r) 1 . S (t,~r) ≡ ∇~vv(t,~r) + ∇~ (II.15b) 2 3 22 Kinematics of a continuous medium Component-wise, and generalizing to curvilinear coordinates, this reads D ij (t,~r) =  1 ~ ∇ · ~v(t,~r) gij (t,~r) + S ij (t,~r) 3 with [cf. Eq (II4a)]      dvk (t,~r) dvl (t,~r) 1 k 2 ~ l + S ij (t,~r) ≡ g (t,~r)g j (t,~r) − ∇ · ~v(t,~r) gij (t,~r) . 2 i 3 dxl dxk (II.15c) (II.15d) Summary Gathering the findings of

this Section, the most general motion of a material volume element inside a continuous medium, in particular in a fluid, can be decomposed in four elements: :::::::::: • a translation; ~ r) given by Eq. (II8)ie related to the anti• a rotation, with a local angular velocity Ω(t,~ symmetric part R (t,~r) of the velocity gradientand equal to twice the (local) vorticity vector ω ~ (t,~r); • a local dilatation or contraction, in which the geometric form of the material volume element ~ · ~v(t,~r), i.e remains unchanged, whose rate is given by the divergence of the velocity field ∇ encoded in the diagonal elements of the strain rate tensor D (t,~r); • a change of shape (“deformation”) of the material volume element at constant volume, controlled by the rate-of-shear tensor S (t,~r) [Eqs. (II15b),(II15d)], obtained by taking the traceless symmetric part of the velocity gradient II.2 Classification of fluid flows The motion, or flow (xxix) , of a fluid can be characterized

according to several criteria, either purely geometrical (Sec. II21), kinematic (Sec II22), or of a more physical nature (Sec II23), ie taking into account the physical properties of the flowing fluid. II.21 Geometrical criteria In the general case, the quantities characterizing the properties of a fluid flow will depend on time as well as on three spatial coordinates. For some more or less idealized models of actual flows, it may turn out that only two spatial coordinates play a role, in which case one talks of a two-dimensional flow . An example is the flow of air around the wing of an airplane, which in first approximation is “infinitely” long compared to its transverse profile: the (important!) effects at the ends of the wing, which introduce the dependence on the spatial dimension along the wing, may be left aside in a first approach, then considered in a second, more detailed step. In some cases, e.g for fluid flows in pipes, one may even assume that the properties only

depend on a single spatial coordinate, so that the flow is one-dimensional . In that approximation, the physical local quantities are actually often replaced by their average value over the cross section of the pipe. On a different level, one also distinguishes between internal und external fluid flows, according to whether the fluid is enclosed inside solid wallse.g in a pipeor flowing around a bodyeg around an airplane wing. (xxix) Strömung 23 II.2 Classification of fluid flows II.22 Kinematic criteria The notions of uniformthat is independent of positionand steadyindependent of time motions were already introduced at the end of Sec. I33 Accordingly, there are non-uniform and unsteady fluids flows. If the vorticity vector ω ~ (t,~r) vanishes at every point ~r of a flowing fluid, then the corresponding motion is referred to as an irrotational flow (xxx) or, for reasons that will be clarified in Sec. IV4, potential flow . The opposite case is that of a vortical or rotational

flow (xxxi) According to whether the flow velocity v is smaller or larger than the (local) speed of sound cs , one talks of subsonic or supersonic motion(xxxii) , corresponding respectively to a dimensionless Mach number (e) v Ma ≡ (II.16) cs smaller or larger than 1. Note that the Mach number can a priori be defined, and take different values Ma(t,~r), at every point in a flow. When the fluid flows in layers that do not mix with each other, so that the streamlines remain parallel, the flow is referred to as laminar . In the opposite case the flow is turbulent II.23 Physical criteria All fluids are compressible, more or less according to the substance and its thermodynamic state. Nevertheless, this compressibility is sometimes irrelevant for a given motion, in which case it may fruitful to consider that the fluid flow is incompressible, which, as seen in § II.13 a, technically ~ ·~v = 0. In the opposite case (∇ ~ ·~v 6= 0), the flow means that its volume expansion rate

vanishes, ∇ is said to be compressible. It is however important to realize that the statement is more a kinematic one, than really reflecting the thermodynamic compressibility of the fluid. In practice, flows are compressible in regions where the fluid velocity is “large”, namely where the Mach number (II.16) is not much smaller than 1, ie roughly speaking Ma & 02 In an analogous manner, one speaks of viscous resp. non-viscous flows to express the fact that the fluid under consideration is modeled as viscous resp. inviscidwhich leads to different equations of motion, irrespective of the fact that every fluid has a non-zero viscosity. Other thermodynamic criteria are also used to characterize possible fluid motions: isothermal flowsi.e in which the temperature is uniform and remains constant, isentropic flowsie without production of entropy, and so on Bibliography for Chapter II • National Committee for Fluid Mechanics film & film notes on Deformation of Continuous

Media; • Faber [1] Chapter 2.4; • Feynman [8, 9] Chapter 39–1; • Guyon et al. [2] Chapters 31, 32; • Sommerfeld [5, 6] Chapter I. (xxx) (e) wirbelfreie Strömung E. Mach, 1838–1916 (xxxi) Wirbelströmung (xxxii) Unterschall- bzw. Überschallströmung Appendix to Chapter II II.A Deformations in a continuous medium Strain tensor. C HAPTER III Fundamental equations of non-relativistic fluid dynamics Some of the most fundamental laws of physics are conservation equations for various quantities: energy, momentum, (electric) charge, and so on. When applying them to many-body systems, in particular to continuous media like moving fluids, care must be taken to consider isolated and closed systems, to ensure their validity. At the very least, the amount of quantity exchanged with the exterior of the systemfor example the change in momentum per unit time due to external forces, as given by Newton’s second law, or the change in energy due to the mechanical work of

these forcesmust be quantifiable. When this is the case, it is possible to re-express global conservation laws or more generally balance equationsgiven in terms of macroscopic quantities like total mass, total energy, total momentum, etc.in a local form involving densities, using the generic recipe provided by Reynolds’ transport theorem (Sec. III1) In the framework of a non-relativistic theory, in which the mass or equivalently the particle number of a closed system is conserved, one may thus derive a general continuity equation, holding at every point of the continuous medium (Sec. III2) The same approach may be followed to derive equations expressing the time evolution of momentum or energy under the influence of external forces acting at every point of the fluid. In either case, it is however necessary to account for the possibility that several physical phenomena may contribute to the transport of momentum and energy. Depending on whether or not, and how, every form of transport

is taken into account, one has different fluid models, leading to different equations for the local expressions of Newton’s second law (Sec. III3) or of energy balance (Sec III4) III.1 Reynolds transport theorem The material derivative of a quantity was already introduced in Sec. I34, in which its action on a local function of both time t and position vector ~r was given. In this Section, we shall derive a formula for the substantial derivative of an extensive physical quantity attached to a “macroscopic” material system. This formula will in the remainder of the Chapter represent the key relation which will allow us to express the usual conservation laws of mechanics, which hold for closed systems, in terms of Eulerian variables. III.11 Closed system, open system Consider the motion of a continuous medium, in particular a flowing fluid, in a reference frame R. Let S be an arbitrary closed geometrical surface, which remains fixed in R This surface will hereafter be referred to

as control surface, and the geometrical volume V it encloses as control volume. Due to the macroscopic transport of matter in the flowing medium, the fluid contained inside the control surface S represents an open system, which can exchange matter with its exterior as time elapses. Let Σ be the closed system consisting of the material points that occupy the control volume V at some given time t. At a shortly later time t + δt, the material system Σ has moved and 26 Fundamental equations of non-relativistic fluid dynamics boundary of Σ at time t  boundary of Σ at time t + δt 2+ - 1 2- streamlines Figure III.1 – Time evolution of a closed material system transported in the motion of a continuous medium. it occupies a new volume in the reference frame. On Fig III1, one can distinguish between three regions in position space: • (1), which is common to the successive positions of Σ at t ant t + δt; • (2−), which was left behind by Σ between t and t + δt; •

(2+), into which Σ penetrates between t and t + δt. III.12 Material derivative of an extensive quantity Let G (t) be one of the extensive quantities that characterize a macroscopic physical property of the closed material system Σ. To this extensive quantity, one can associate at every point ~r the corresponding intensive specific density gm(t,~r), defined as the local amount of G per unit mass of matter. Denoting by dG (t,~r) resp dM (t,~r) the amount of G resp the mass inside a small material volume at position ~r at time t, one can write symbolically gm(t,~r) = dG (t,~r) , dM (t,~r) (III.1) where the notation with differentials was used to suggest that the identity holds in the limit of a small material volume. For instance, the linear momentum resp. the kinetic energy of a mass dM of fluid moving with ~ velocity ~v is dP~ =~v dM resp. dK =~v2 dM/2, so that the associated specific density is dP/dM =~v 2 resp. dK/dM =~v /2 Remark: These examples illustrate the fact that the

tensorial naturescalar, vector, tensor of higher rankof the function associated with quantity G can be arbitrary. For a material system Σ occupying at time t a volume V bounded by the control surface S, Eq. III1 leads to Z Z G (t) = g m(t,~r) dM = gm(t,~r) ρ(t,~r) d3~r (III.2) V V for the value of G of the system, where in the second identity ρ(t,~r) = dM/d3~r is the local mass density. Let us now assume that the material system Σ is moving as part of a larger, flowing continuous medium. To find the substantial derivative DG (t)/Dt of G (t), we shall first compute the variation 27 III.1 Reynolds transport theorem δ G for the material system Σ between times t and t + δt, where δ is assumed to be small. At the end of the calculation, we shall take the limit δt 0. Going back to the regions (1), (2−), (2+) defined in Fig. III1, one can write

δ G = G 1 + G 2+ t+δt − G 1 + G 2− t = δ G 1 + δ G 2 , where the various indices denote the respective spatial

domains and instants, and



δ G 2 ≡ G 2+ t+δt − G 2− t . δ G 1 ≡ G 1 t+δt − G 1 t , • δ G 1 represents the variation of G inside region (1) due to the non-stationarity of the fluid flow. In the limit δt 0, this region (1) coincides with the control volume V: to leading order in δt, one thus has  Z Z  dG 1 (t) ∂ d 3 δG 1 = gm(t,~r) ρ(t,~r) d ~r δt = gm(t,~r) ρ(t,~r) d3~r δt, (III.3) δt = dt dt V V ∂t where the first identity is a trivial Taylor expansion, the second one replaces the volume of region (1) by V, while the last identity follows from the independence of the control volume from time. • δ G 2 represents the algebraic amount of G traversing between t and t+δt the control surface S, either leaving (region 2+) or entering (region 2−) the control volume V, where in the latter case the amount is counted negatively. This is precisely the flux in the mathematical acceptation of the termthrough the surface S, oriented towards the exterior,

of an appropriate flux density for quantity G .(2)   d2 S  |~v| δt : ~v    - Let ~v denote the velocity of the continuous medium at position ~r at time t. The amount of quantity G that traverses in δt a surface element d2 S situated in ~r equals the amount inside an elementary cylinder with base d2 S and height |~v| δt, i.e d3 G = gm ρ d3 V , with ~ is normal to the surface ~ ·~v| δt, where the vector d2 S d3 V = |d2 S element. Integrating over all surface elements all over the control surface, the amount of quantity G flowing through S thus reads I I   ~ δt. δG 2 = d3 G = (III.4) gm(t,~r) ρ(t,~r)~v(t,~r) · d2 S S S All in all, Eqs. (III3)–(III4) yield after dividing by δt and taking the limit δt 0 the so-called Reynolds transport theorem:(xxxiii)(f) DG (t) = Dt Z  ∂ gm(t,~r) ρ(t,~r) d3~r + V ∂t I S ~ gm(t,~r) ρ(t,~r)~v(t,~r) · d2 S.   (III.5) The first term on the right hand side of this relation represents a local time derivative

∂ G /∂t, similar to the first term in Eq. I18 In contrast, the second term is of convective type, ie directly caused by the motion of matter, and represents the transport of G . (2) This flux density can be read off Eq. (III4), namely (xxxiii) (f) Reynolds’scher Transportsatz O. Reynolds, 1842–1912 gm(t,~r) ρ(t,~r)~v(t,~r). 28 Fundamental equations of non-relativistic fluid dynamics Anticipating on the rest of the Chapter, this theorem will help us as follows. The “usual” laws of dynamics are valid for closed, material systems Σ, rather than for open ones. Accordingly, these laws involve time derivatives “following the system in its motion”, that is precisely the material derivative D/Dt. Reynolds’ transport theorem (III5) expresses the latter, for extensive quantities G (t), in terms of local densities attached to fixed spatial positions, i.e in Eulerian variables Remarks: ∗ The medium velocity ~v(t,~r) entering Reynolds transport theorem (III.5) is

measured in the reference frame R in which the control surface S remains motionless. ∗ Since relation (III.5) is traditionally referred to as a theorem, one may wonder what are its assumptions. Obviously, the derivation of the result relies on the assumption that the specific density gm(t,~r) and the velocity field ~v(t,~r) are both continuous and differentiable, in agreement with the generic hypotheses in Sec. I22 Figure III1 actually also embodies the hidden, but necessary assumption that the motion is continuous, which leads to the smooth evolution of the connected system of material points which are together inside the control surface S at time t. Again, this follows from suitable properties of ~v. ∗ Accordingly, the Reynolds transport theorem (III.5) does not hold if the velocity field, or the specific density gm, is discontinuous. As was already mentioned in Sec I22, such discontinuities are however necessary to account for some phenomena (shock waves, boundary between two

immiscible fluids. ) In such cases, it will be necessary to reformulate the transport theorem to take into account the discontinuities. III.2 Mass and particle number conservation: continuity equation The mass M and the particle number N of a closed non-relativistic system Σ remain constant in its motion: DN (t) DM (t) = 0, = 0. (III.6) Dt Dt These conservation laws lead with the help of Reynolds’ transport theorem to partial differential equations for some of the local fields that characterize a fluid flow. III.21 Integral formulation For an arbitrary control volume V delimited by surface S, the Reynolds transport theorem (III.5) with G (t) = M , to which corresponds the specific density gm(t,~r) = 1, reads  I Z   DM (t) ∂ ~ = 0. ρ(t,~r) d3~r + (III.7) ρ(t,~r)~v(t,~r) · d2 S = Dt ∂t V S That is, the time derivative of the mass contained in V is the negative of the mass flow rate through S. In agreement with footnote 2, ρ(t,~r)~v(t,~r) is the mass flux

density,(xxxiv) while its integral is the mass flow rate.(xxxv) Taking now G (t) = N , the associated specific density is gm(t,~r) = N/M . Since the product of N/M with the mass density ρ(t,~r) is precisely the particle number density n (t,~r), Reynolds’ theorem (III.5) leads to  I Z   ∂ DN (t) 3 ~ = 0, (III.8) n (t,~r) d ~r + n (t,~r)~v(t,~r) · d2 S = Dt ∂t V S where n (t,~r)~v(t,~r) represents the particle number flux density.(xxxvi) (xxxiv) Massenstromdichte (xxxv) Massenstrom (xxxvi) Teilchenstromdichte III.3 Momentum balance: Euler and Navier–Stokes equations 29 Equation (III.7) resp (III8) consitutes the integral formulation of mass resp particle number conservation. Remark: In the case of a steady motion, Eq. (III7) shows that the net mass flow rate through an arbitrary closed geometrical surface S vanishes. That is, the entrance of some amount of fluid into a (control) volume V must be compensated by the simultaneous departure of an equal mass from the

volume. III.22 Local formulation Since the control volume V in Eq. (III7) resp (III8) is time-independent, the time derivative can be exchanged with the integration over volume. Besides, the surface integral can be transformed with the help of the Gauss theorem into a volume integral. All in all, this yields  Z   3 ∂ρ(t,~r) ~  + ∇ · ρ(t,~r)~v(t,~r) d ~r = 0, ∂t V resp. Z  V   3 ∂ n (t,~r) ~  + ∇ · n (t,~r)~v(t,~r) d ~r = 0. ∂t These identities hold for an arbitrary integration volume V. Using the continuity of the respective integrands, one deduces the following so-called continuity equations:  ∂ρ(t,~r) ~  + ∇ · ρ(t,~r)~v(t,~r) = 0 ∂t (III.9) resp.  ∂ n (t,~r) ~  + ∇ · n (t,~r)~v(t,~r) = 0. (III.10) ∂t Equation (III.9) represents the first of five dynamical (partial differential) equations which govern the evolution of a non-relativistic fluid flow. Remarks: ∗ The form of the continuity equation (III.9) does not depend on the

properties of the flowing medium, as for instance whether dissipative effects play a significant role or not. This should be contrasted with the findings of the next two Sections.   ~ · ρ(t,~r)~v(t,~r) = 0, i.e ∗ In the case of a steady fluid flow, Eq. (III9) gives ∇ ~ · ~v(t,~r) +~v(t,~r) · ∇ρ(t,~ ~ ρ(t,~r) ∇ r) = 0. Thus, the stationary flow of a homogeneous fluid, i.e for which ρ(t,~r) is position independent, will ~ ·~v(t,~r) = 0, cf. Eq (II13)] be incompressible [∇ III.3 Momentum balance: Euler and Navier–Stokes equations For a closed system Σ with total linear momentum P~ with respect to a given reference frame R, Newton’s second law reads DP~ (t) = F~ (t), (III.11) Dt with F~ the sum of the “external” forces acting on Σ. The left hand side of this equation can be transformed with the help of Reynolds’ transport theorem (III.5), irrespective of any assumption on the fluid under consideration (Sec III31) In contrast, the forces acting on a fluid

element, more precisely the forces exerted by the neighboring 30 Fundamental equations of non-relativistic fluid dynamics elements, do depend on the properties of the fluid. The two most widespread models used for fluids are that of a perfect fluid, which leads to the Euler equation (Sec. III32), and of a Newtonian fluid, for which Newton’s second law (III.11) translates into the Navier–Stokes equation (Sec III33) Throughout this Section, we use the shorter designation “momentum” instead of the more accurate “linear momentum”. III.31 Material derivative of momentum As already noted shortly below Eq. (III1), the specific density associated with the momentum ~ P (t) is simply the flow velocity ~v(t,~r). Applying Reynolds’ theorem (III5) for the momentum of the material system contained at time t inside a control volume V, the material derivative on the left hand side of Newton’s law (III.11) can be recast as  I Z DP~ (t) ∂ 3 ~ ~v(t,~r) ρ(t,~r) d ~r + ~v(t,~r)

ρ(t,~r)~v(t,~r) · d2 S. (III.12) = Dt ∂t V S Both terms on the right hand side can be transformed, to yield more tractable expressions. On the one hand, since the control volume V is immobile in the reference frame R, the time derivative can be taken inside the integral. Its action on ρ(t,~r)~v(t,~r) is then given by the usual product rule On the other hand, one can show the identity  I Z    ∂ρ(t,~r) 2~ ~ ~v(t,~r) ρ(t,~r)~v(t,~r) · d S = + ρ(t,~r) ~v(t,~r) · ∇ ~v(t,~r) d3~r. (III.13) −~v(t,~r) ∂t S V All in all, one thus obtains   Z Z  DP~ (t) D~v(t,~r) 3 ∂~v(t,~r)  3 ~ = ρ(t,~r) + ~v(t,~r) · ∇ ~v(t,~r) d ~r = ρ(t,~r) d ~r. Dt ∂t Dt V V (III.14) ~ denote the vector defined by the surface integral on the left Proof of relation (III.13): let J(t) hand side of that identity. For the i-th component of this vector, Gauss’ integral theorem gives I Z  i    ~ = ∇ ~ · vi (t,~r) ρ(t,~r)~v(t,~r) d3~r. J i (t) = v (t,~r) ρ(t,~r)~v(t,~r) · d2 S S V

  ~ · ρ(t,~r)~v(t,~r) + ρ(t,~r)~v(t,~r) · ∇v ~ i (t,~r): The action of the differential operator yields v (t,~r) ∇ the divergence in the first term can be expressed according to the continuity equation (III.9) as the negative of the time derivative of the mass density, leading to     ~ vi (t,~r). ~ · vi (t,~r) ρ(t,~r)~v(t,~r) = −vi (t,~r) ∂ρ(t,~r) + ρ(t,~r) ~v(t,~r) · ∇ ∇ ∂t ~ from where Eq. (III13) follows This relation holds for all three components of J,  i Remark: The derivation of Eq. (III14) relies on purely algebraic transformations, either as encoded in Reynolds’ transport theorem, or when going from relation (III.12) to (III14) That is, it does not imply any modelapart from that of a continuous mediumfor the fluid properties. In particular, whether or not dissipative effects are important in the fluid did not play any role here. III.32 Perfect fluid: Euler equation In this section, we first introduce the notion, or rather the model, of a

perfect fluid , which is defined by the choice of a specific ansatz for the stress tensor which encodes the contact forces between neighboring fluid elements. Using that ansatz and the results of the previous paragraph, Newton’s second law (III.11) is shown to be equivalent to a local formulation, the so-called Euler equation. Eventually, the latter is recast in the generic form for a local conservation or balance equation, involving the time derivative of a local density and the divergence of the corresponding flux density. 31 III.3 Momentum balance: Euler and Navier–Stokes equations III.32 a Forces in a perfect fluid :::::::::::::::::::::::::::::::: The forces in a fluid were already discussed on a general level in Sec. I41 Thus, the total force on the right hand side of Eq. (III11) consist of volume and surface forces, which can respectively be expressed as a volume or a surface integral Z I F~ (t) = f~V (t,~r) d3~r + T~s (t,~r) d2 S, (III.15) V S where f~V denotes the

local density of body forces, while T~s is the mechanical stress vector introduced in Eq. (I20) The latter will now allow us to introduce various models of fluids The first, simplest model is that of a perfect fluid , or ideal fluid : A perfect fluid is a fluid in which there are no shear stresses nor heat conduction. (III.16a) Stated differently, at every point of a perfect fluid the stress vector T~s on a (test) surface element moving with the fluid is normal to d2 S, irrespective of whether the fluid is at rest or in motion. That is, introducing the normal unit vector ~en (~r) to d2 S oriented towards the exterior of the material region acted upon,(3) one may write d2 S T~s (t,~r) = −P (t,~r)~en (~r), (III.16b) with P (t,~r) the pressure at position ~r. Accordingly, the mechanical stress tensor in a perfect fluid in a reference frame R which is moving with the fluid is given by σ (t,~r) = −P (t,~r) g−1 (t,~r), (III.16c) with g−1 the inverse
metric tensor, just like

in a fluid at rest [Eq. (I23)] In a given coordinate system in R, the 20 -components of σ thus simply read σ ij (t,~r) = −P (t,~r) g ij (t,~r) (III.16d)
i.e the 11 -components are σ ij (t,~r) = −P (t,~r) δ ij Using relation (III.16b), the total surface forces in Eq (III15) can be transformed into a volume integral: Z I I I 2 2 2 ~ P (t,~r) d3~r, ~ =− ∇ (III.17) T~s (t,~r) d S = − P (t,~r)~en (~r) d S = − P (t,~r) d S S S S V where the last identity follows from a corollary of the usual divergence theorem. P (t,~r) entering Eqs. (III16b)– (III.16d) is actually the hydrostatic pressure already introduced in the definition of the mechanical stress in a fluid at rest, see Eq. (I23) One heuristic justification is that the stresses are defined as the forces per unit area exerted by a piece of fluid situated on one side of a surface on the fluid situated on the other side. Even if the fluid is moving, the two fluid elements on both sides of the surfaceas well as

the comoving test surfacehave the same velocity,(4) i.e their relative velocity vanishes, just like in a fluid at rest. Remark: Although this might not be intuitive at first, the pressure III.32 b Euler equation ::::::::::::::::::::::: Gathering Eqs. (III11), (III14), (III15) and (III17) yields   Z h Z i  ∂~v(t,~r)  3 ~ ~ P (t,~r) + f~V (t,~r) d3~r. + ~v(t,~r) · ∇ ~v(t,~r) d ~r = −∇ ρ(t,~r) ∂t V V (3) (4) Cf. the discussion between Eqs (I21a)–(I21c) . thanks to the usual continuity assumption: this no longer holds at a discontinuity! 32 Fundamental equations of non-relativistic fluid dynamics Since this identity must hold irrespective of the control volume V, the integrands on both sides must be equal. That is, the various fields they involve obey the Euler equation    ∂~v(t,~r)  ~ P (t,~r) + f~V (t,~r). ~ ρ(t,~r) + ~v(t,~r) · ∇ ~v(t,~r) = −∇ ∂t (III.18) Remarks: ∗ The term in curly brackets on the left hand side is exactly the

acceleration (I.17) of a material point, as in Newton’s second law. ~ v, the Euler equation is a non-linear partial differential equation. ∗ Due to the convective term (~v ·∇)~ ∗ Besides Newton’s second law for linear momentum, one could also think of investigating the consequence of its analogue for angular momentum. Since we have assumed that the material points do not have any intrinsic spin, the conservation of angular momentum, apart from leading to the necessary symmetry of the stress tensorwhich is realized in a perfect fluid, see Eq. (III16c) or (III.16d), and will also hold in a Newtonian fluid, see Eq (III26)does not bring any new dynamical equation. III.32 c Boundary conditions ::::::::::::::::::::::::::::: To fully formulate the mathematical problem representing a given fluid flow, one must also specify boundary conditions for the various partial differential equations. These conditions reflect the geometry of the problem under consideration. • Far from an

obstacle or from walls, one may specify a given pattern for the flow velocity field. For instance, one may require that the flow be uniform “at infinity”, as e.g for the motion far from the rotating cylinder in Fig. IV5 illustrating the geometry of the Magnus effect • At an obstacle, in particular at a wall, the component of velocity perpendicular to the obstacle should vanish: that is, the fluid cannot penetrate the obstacle or wall, which makes sense and will be hereafter often referred to as impermeability condition. In case the obstacle is itself in motion, one should consider the (normal component of the) relative velocity of the fluid with respect to the obstacle. On the other hand, the model of a perfect fluid, in which there is by definition no friction, does not specify the value of the tangential component of the fluid relative velocity at an obstacle. III.32 d Alternative forms of the Euler equation ::::::::::::::::::::::::::::::::::::::::::::::: In practice, the

volume forces acting on a fluid element are often proportional to its mass, as are e.g the gravitational, Coriolis or centrifugal forces Therefore, it is rather natural to introduce the corresponding force density per unit mass, instead of per unit volume: dF~V (t,~r) f~V (t,~r) ~aV (t,~r) ≡ = . dM (t,~r) ρ(t,~r) With the help of this “specific density of body forces”, which has the dimension of an acceleration, the Euler equation (III.18) can be recast as 1 ~ D~v(t,~r) =− ∇P (t,~r) + ~aV (t,~r). Dt ρ(t,~r) (III.19) The interpretation of this form is quite straightforward: the acceleration of a material point (left hand side) is the sum of the acceleration due to the pressure forces and the acceleration due to volume forces (right hand side). III.3 Momentum balance: Euler and Navier–Stokes equations 33 Alternatively, one may use the identity (in which the time and position variables have been omitted for the sake of brevity)  2

~v ~ ~ ~ ~v, ~v × ∇ ×~v =

∇ − ~v · ∇ 2 which can be proved either starting from the usual formula for the double cross productwith a small twist when applying the differential operator to a vector squaredor by working component by component. Recognizing in the rightmost term the convective part of the Euler equation, one can rewrite the latter, or equivalently Eq. (III19), as   1 ~ ∂~v(t,~r) ~ ~v(t,~r)2 − ~v(t,~r) × ω ~ (t,~r) = − ∇P (t,~r) + ~aV (t,~r), +∇ ∂t 2 ρ(t,~r) (III.20) where we have made use of the vorticity vector (II.10) Note that the second term on the left hand side of this equation involves the (gradient of the) kinetic energy per unit mass dK/dM . In Sec. IV21, we shall see yet another form of the Euler equation [Eq (IV8)], involving thermodynamic functions other than the pressure III.32 e The Euler equation as a balance equation The Euler equation can be rewritten in the generic form for of a balance equation, namely as the identity of the sum of the time derivative of

a density and the divergence of a flux density with a source termwhich vanishes if the quantity under consideration is conserved. Accordingly, we first introduce two :::::::::::::::::::::::::::::::::::::::::::::::::: Definitions: One associates with the i-th component in a given coordinate system of the momentum of a material system its • density (xxxvii) ρ(t,~r) vi (t,~r) and (III.21a) • flux density (xxxviii) (in direction j) T ij (t,~r) ≡ P (t,~r) g ij (t,~r) + ρ(t,~r) vi (t,~r) vj (t,~r), (III.21b) with g ij the components of the inverse metric tensor g−1 . Physically, T ij represents the amount of momentum along ~ei transported per unit time through a unit surface(5) perpendicular to the direction of ~ej i.e transported in direction j That is, it is the i-th component of the force upon a test unit surface with normal unit vector ~ej . The first contribution to T ij , involving pressure, is the transport due to the thermal, random motion of the atoms of the fluid.

On the other hand, the second termnamely the transported momentum density multiplied by the velocityarises from the convective transport represented by the macroscopic motion. Remarks: ∗ As thermal motion is random and (statistically) isotropic, it does not contribute to the momentum density ρ(t,~r)~v(t,~r), only to the momentum flux density.
∗ In tensor notation, the momentum flux density (III.21b), viewed as a 20 -tensor, is given by T (t,~r) = P (t,~r)g−1 (t,~r) + ρ(t,~r)~v(t,~r) ⊗~v(t,~r) for a perfect fluid. (5) (III.22) . which must be immobile in the reference frame in which the fluid has the velocity ~v entering definition (III21b) (xxxvii) Impulsdichte (xxxviii) Impulsstromdichte 34 Fundamental equations of non-relativistic fluid dynamics ∗ Given its physical meaning, the momentum flux (density) tensor T is obviously related to the Cauchy stress tensor σ . More precisely, T represents the forces exerted by a material point on its neighbors, while σ

stands for the stresses acting upon the material point due to its neighbors. Invoking Newton’s third lawwhich in continuum mechanics is referred to as Cauchy’s fundamental lemma, these two tensors are simply opposite to each other. ∗ Building on the previous remark, the absence of shear stress defining a perfect fluid can be reformulated as a condition of the momentum flux tensor: A perfect fluid is a fluid at each point of which one can find a local velocity, such that for an observer moving with that velocity the fluid is locally isotropic. (III.23) The momentum flux tensor is thus diagonal in the observer’s reference frame. Using definitions (III.21), one easily checks that the Euler equation (III18) is equivalent to the balance equations (for i = 1, 2, 3) 3  X dT Tij (t,~r) ∂ ρ(t,~r) vi (t,~r) + = fVi (t,~r). ∂t dxj (III.24a) j=1 with fVi the i-th component of the volume force density and d /dxi the covariant derivatives (see Appendix C.1), that coincide with the

partial derivatives in Cartesian coordinates Proof: For the sake of brevity, the (t,~r)-dependence of the various fields will not be specified. One finds 3 3 3 3 j i X X Tij ∂(ρvi ) X dT ∂ vi X ij dP ∂ρ i i d(ρv ) j dv + v + ρ + g = + + v ρv ∂t dxj ∂t ∂t dxj j=1 dxj dxj j=1 j=1 j=1   i   ∂v ∂ρ ~ ~ i + dP , + ∇ · (ρ~v) + ρ + (~v · ∇)v = vi ∂t ∂t dxi P ij where we have used j g d/dxj = d/dxi . The first term between square brackets vanishes thanks to the continuity equation (III.9) In turn, the second term is precisely the i-th component of the left member of the Euler equation (III.18), ie it equals the i-th component of f~V minus ~ P. the third term, which represents the i-th component of ∇  In tensor notation, Eq. (III24a) reads  ∂ ~ · T (t,~r) = f~V (t,~r), ρ(t,~r)~v(t,~r) + ∇ ∂t (III.24b) where we
have used the symmetry of the momentum flux tensor T , while the action of the divergence on a 20 -tensor is defined through its

components, which is to be read off Eq. (III24a) III.33 Newtonian fluid: Navier–Stokes equation In a real moving fluid, there are friction forces that contribute to the transport of momentum between neighboring fluid layers when the latter are in relative motion. Accordingly, the momentum flux-density tensor is no longer given by Eq. (III21b) or (III22), but now contains extra terms, involving derivatives of the flow velocity. Accordingly, the Euler equation must be replaced by an alternative dynamical equation, including the friction forces. III.33 a Momentum flux density in a Newtonian fluid ::::::::::::::::::::::::::::::::::::::::::::::::::::: The momentum flux density (III.21b) in a perfect fluid only contains two termsone proportional to the components g ij of the inverse metric tensor, the other proportional to vi (t,~r) vj (t,~r) 35 III.3 Momentum balance: Euler and Navier–Stokes equations Since the coefficients in front of these two terms could a priori depend on~v2

, this represents the most general symmetric tensor of degree 2 which can be constructed with the help of the flow velocity only. If the use of terms that depend on the spatial derivatives of the velocity field is also allowed, the components of the momentum flux-density tensor can be of the following form, where for the sake of brevity the variables t and ~r are omitted ! i j 2v dv dv d i + ··· , (III.25) +B +O T ij = P g ij + ρvi vj + A dxj dxi dxj dxk with coefficients A, B that depend on i, j and on the fluid under consideration. This ansatz for T ij , as well as the form of the energy flux density involved in Eq. (III35) below, can be “justified” by starting from a microscopic kinetic theory of the fluid and writing the solutions of the corresponding equation of motion as a specific expansionwhich turns out to be in powers of the Knudsen number (I.4) This also explains why terms of the type vi ∂ P /∂xj or vi ∂T /∂xj , with T the temperature, were not considered in

Eq. (III25) Despite these theoretical considerations, in the end the actual justification for the choices of momentum or energy flux density is the agreement with the measured properties of fluids. As discussed in Sec. I13, the description of a system of particles as a continuous medium, and in particular as a fluid, in local thermodynamic equilibrium, rests on the assumption that the macroscopic quantities of relevance for the medium vary slowly both in space and time. Accordingly, (spatial) gradients should be small: the third and fourth terms in Eq. (III25) should thus be on the one hand much smaller than the first two ones, on the other hand much larger than the rightmost term as well as those involving higher-order derivatives or of powers of the first derivatives. Neglecting these smaller terms, one obtains “first-order dissipative fluid dynamics”, which describes the motion of Newtonian fluidsthis actually defines the latter. Using the necessary symmetry of T ij , the third

and fourth terms in Eq. (III25) can be rewritten as the sum of a traceless symmetric contribution and a tensor proportional to the inverse metric tensor. This leads to the momentum flux-density tensor T ij (t,~r) = P (t,~r) g ij (t,~r) + ρ(t,~r)vi (t,~r)vj (t,~r)  i  dv (t,~r) dvj (t,~r) 2 ij ~ + − g (t,~r)∇ ·~v(t,~r) − η(t,~r) dxj dxi 3 (III.26a) ~ ·~v(t,~r). − ζ(t,~r)g ij (t,~r)∇ In geometric formulation, this reads T (t,~r) = P (t,~r) g−1 (t,~r) + ρ(t,~r)~v(t,~r) ⊗~v(t,~r) + π (t,~r) (III.26b) where dissipative effects are encored in the viscous stress tensor (xxxix)     −1  1 ~ ~ v(t,~r) g−1 (t,~r) π (t,~r) ≡ −2η(t,~r) D (t,~r)− ∇·~v(t,~r) g (t,~r) −ζ(t,~r) ∇·~ 3 (III.26c) for a Newtonian fluid with D (t,~r) the strain rate tensor discussed in Sec. II13 Component-wise      1 ~ ~ ·~v(t,~r) g ij (t,~r). πij (t,~r) ≡ −2η(t,~r) D ij (t,~r) − ∇ ·~v(t,~r) g ij (t,~r) − ζ(t,~r) ∇ 3 (xxxix) viskoser

Spannungstensor (III.26d) 36 Fundamental equations of non-relativistic fluid dynamics In terms of the traceless rate-of-shear tensor (II.15b) or of its components (II15d), one may alternatively write   ~ ·~v(t,~r) g−1 (t,~r) π (t,~r) ≡ −2η(t,~r) S (t,~r) − ζ(t,~r) ∇ (III.26e)   ~ ·~v(t,~r) g ij (t,~r). πij (t,~r) ≡ −2η(t,~r) S ij (t,~r) − ζ(t,~r) ∇ (III.26f) This viscous stress tensor involves two novel characteristics of the medium, so-called transport coefficients, namely: • the (dynamical) shear viscosity (xl) η, which multiplies the traceless symmetric part of the velocity gradient tensor, i.e the conveniently termed rate-of-shear tensor; • the bulk viscosity, also called second viscosity,(xli) ζ, which multiplies the volume-expansion ~ ·~v(t,~r). part of the velocity gradient tensor, i.e the term proportional to ∇ The two corresponding contributions represent a diffusive transport of momentum in the fluid representing a third type of

transport besides the convective and thermal ones. Remarks: ∗ In the case of a Newtonian fluid, the viscosity coefficients η and ζ are independent of the flow velocity. However, they still depend on the temperature and pressure of the fluid, so that they are not necessarily uniform and constant in a real flowing fluid. ~ · ~v(t,~r) = 0, the last contribution to the momentum flux den∗ In an incompressible flow, ∇ sity (III.26) drops out Thus, the bulk viscosity ζ only plays a role in compressible fluid motions(6) ∗ Expression (III.26c) or (III26d) of the viscous stress tensor assumes implicitly that the fluid is (locally) isotropic, since the coefficients η, ζ are independent of the directions i, j. III.33 b Surface forces in a Newtonian fluid :::::::::::::::::::::::::::::::::::::::::::: The Cauchy stress tensor corresponding to the momentum flux density (III.26) of a Newtonian fluid is σ (t,~r) = −P (t,~r) g−1 (t,~r) − π (t,~r) (III.27a) that is, using the form

(III.26e) of the viscous stress tensor   ~ ·~v(t,~r) g−1 (t,~r). σ (t,~r) = −P (t,~r) g−1 (t,~r) + 2η(t,~r) S (t,~r) + ζ(t,~r) ∇ (III.27b) Component-wise, this becomes       i 2 dv (t,~r) dvj (t,~r) ij ij ~ σ (t,~r) = − P (t,~r)+ ζ(t,~r)− η(t,~r) ∇·~v(t,~r) g (t,~r)+η(t,~r) . (III27c) + 3 dxj dxi Accordingly, the mechanical stress vector on an infinitesimally small surface element situated at point ~r with unit normal vector ~en (~r) reads    3  X 2 ~ ~ Ts (t,~r) = σ (t,~r) · ~en (~r) = − P (t,~r) + ζ(t,~r) − η(t,~r) ∇ ·~v(t,~r) g ij (t,~r) 3 i,j=1  i  dv (t,~r) dvj (t,~r) + η(t,~r) nj (~r)~ei (t,~r), (III.28) + dxj dxi with nj (~r) the coordinate of ~en (~r) along direction j. One easily identifies the two components of (6) As a consequence, the bulk viscosity is often hard to measureone has to devise a compressible flowso that it is actually not so well known for many substances, even well-studied ones [11]. (xl)

Scherviskosität (xli) Dehnviskosität, Volumenviskosität, zweite Viskosität 37 III.3 Momentum balance: Euler and Navier–Stokes equations this stress vector (cf. Sec I41) P i j • the term proportional to g j n ~ei = ~en is the normal stress on the surface element. It consists of the usual hydrostatic pressure term −P ~en , and a second one proportional to the ~ ·~v: in the compressible motion of a Newtonianand more generally a local expansion rate ∇ dissipativefluid, the normal stress is thus not only given by −P ~en , but includes additional contributions that vanish in the static case. • the remaining term is the tangential stress, proportional to the shear viscosity η. Accordingly, the value of the latter can be deduced from a measurement of the tangential force acting on a surface element, see Sec. VI12 As in § III.32 a, the external contact forces acting on a fluid element delimited by a surface S can easily be computed. Invoking the Stokes theorem yields

I I I T~s (t,~r) d2 S = − P (t,~r)~en (~r) d2 S − π (t,~r) · ~en (~r) d2 S S ZS Z S 3 ~ P (t,~r) d V + ∇ ~ · π(t,~r) d3 V =− ∇ ZV ZV ~ P (t,~r) d3 V + f~visc (t,~r) d3 V , =− ∇ (III.29a) V V with the local viscous friction force density   i  3 j (t,~ X d dv r) dv (t,~ r) f~visc (t,~r) = η(t,~r) ~ej (t,~r) + dxi dxj dxi i,j=1    2 ~ ~ + ∇ ζ(t,~r) − η(t,~r) ∇ ·~v(t,~r) . 3 (III.29b) III.33 c Navier–Stokes equation :::::::::::::::::::::::::::::::: Combining the viscous force (III.29b) with the generic equations (III12), (III14) and (III15), the application of Newton’s second law to a volume V of fluid leads to an identity between sums of volume integrals. Since this relation holds for any volume V , it translates into an identity between the integrands, namely    ∂~v(t,~r)  ~ P (t,~r) + f~visc (t,~r) + f~V (t,~r) ~ ~v(t,~r) = −∇ ρ(t,~r) (III.30a) + ~v(t,~r) · ∇ ∂t or component-wise   i     i 2 dP (t,~r) d ∂v (t,~r)  ~

~ ζ(t,~r) − η(t,~r) ∇ ·~v(t,~r) ρ(t,~r) + ~v(t,~r) · ∇ v (t,~r) = − + ∂t dxi dxi 3    3 X d   dvi (t,~r) dvj (t,~r) ~V (t,~r) i + + + f η(t,~ r) dxj dxj dxi j=1 (III.30b) for i = 1, 2, 3. If the implicit dependence of the viscosity coefficients on time and position is negligible, one may pull η and ζ outside of the spatial derivatives. As a result, one obtains the (compressible) Navier–Stokes equation (g),(h)        ∂~v(t,~r)  ~ P (t,~r) + η4~v(t,~r) + ζ + η ∇ ~ ~v(t,~r) = −∇ ~ ·~v(t,~r) + f~V (t,~r) ~ ∇ ρ(t,~r) + ~v(t,~r) · ∇ ∂t 3 (III.31) (g) C.-L Navier, 1785–1836 (h) G. G Stokes, 1819–1903 38 Fundamental equations of non-relativistic fluid dynamics ~ 2 the Laplacian. This is a non-linear partial differential equation of second order, while with 4 = ∇ the Euler equation (III.18) is of first order The difference between the order of the equations is not a mere detail: while the Euler equation looks like the limit

η, ζ 0 of the Navier–Stokes equation, the corresponding is not necessarily true of their solutions. This is for instance due to the fact that their respective boundary conditions differ. In the case of an incompressible flow, the local expansion rate in the Navier–Stokes equation (III.31) vanishes, leading to the incompressible Navier–Stokes equation  ∂~v(t,~r)  ~ P (t,~r) + ν4~v(t,~r), ~ ~v(t,~r) = − 1 ∇ + ~v(t,~r) · ∇ ∂t ρ (III.32) with ν ≡ η/ρ the kinematic shear viscosity. Remark: The dimension of the dynamic viscosity coefficients η, ζ is ML−1 T−1 and the corresponding unit in the SI system is the Poiseuille(i) , abbreviated Pa·s. In contrast, the kinematic viscosity has dimension L2 T−1 , i.e depends only on space and time, hence its denomination III.33 d Boundary conditions ::::::::::::::::::::::::::::: At the interface between a viscous fluid, in particular a Newtonian one, and another bodybe it an obstacle in the flow, a wall

containing the fluid, or even a second viscous fluid which is immiscible with the first onethe relative velocity between the fluid and the body must vanish. This holds not only for the normal component of the velocity (“impermeability” condition), as in perfect fluids, but also for the tangential one, to account for the friction forces. The latter requirement is often referred to as no-slip condition. III.34 Higher-order dissipative fluid dynamics Instead of considering only the first spatial derivatives of the velocity field in the momentum flux-density tensor (III.25), one may wish to also include the second derivatives, or even higher ones. Such assumptions lead to partial differential equations of motion, replacing the Navier–Stokes equation, of increasing order: Burnett equation, super Burnett equation [12]. The domain of validity of such higher-order dissipative fluid models is a priori larger than that of first-order fluid dynamics, since it becomes possible to account

for stronger gradients. On the other hand, this is at the cost of introducing a large number of new parameters besides the transport coefficients already present in Newtonian fluids. In parallel, the numerical implementation of the model becomes more involved, so that a macroscopic description does not necessarily represent the best approach. III.4 Energy conservation, entropy balance The conservation of mass and Newton’s second law for linear momentum lead to four partial differential equations, one scalarcontinuity equation (III.9)and one vectorialEuler (III18) or Navier–Stokes (III.31), describing the coupled evolutions of five fields: ρ(t,~r), the three components of ~v(t,~r) and P (t,~r)(7) To fully determine the latter, a fifth equation is needed For this last constraint, there are several possibilities. A first alternative is if some of the kinematic properties of the fluid flow are known a priori. Thus, requiring that the motion be steady or irrotational or

incompressible. might suffice to fully (7) (i) The density of volume forces f~V or equivalently the corresponding potential energy per unit mass Φ, which stand for gravity or inertial forces, are given “from the outside” and not counted as a degree of freedom. J.-L-M Poiseuille, 1797–1869 III.4 Energy conservation, entropy balance 39 constrain the fluid flow for the geometry under consideration: we shall see several examples in the next three Chapters. A second possibility, which will also be illustrated in Chap. IV–VI, is that of a thermodynamic constraint: isothermal flow, isentropic flow. For instance, one sees in thermodynamics that in an adiabatic process for an ideal gas, the pressure and volume of the latter obey the relation PV γ = constant, where γ denotes the ratio of the heat capacities at constant pressure (CP ) and constant volume (CV ). Since V is proportional to 1/ρ, this so-called “adiabatic equation of state” provides the needed constraint

relating pressure and mass density. Eventually, one may argue that non-relativistic physics automatically implies a further conservation law besides those for mass and linear momentum, namely energy conservation. Thus, using the reasoning adopted in Secs. (III2) and (III3), the rate of change of the total energyinternal, kinetic and potentialof the matter inside a given volume equals the negative of the flow of energy through the surface delimiting this volume. In agreement with the first law of thermodynamics, one must take into account in the energy exchanged with the exterior of the volume not only the convective transport of internal, kinetic and potential energies, but also the exchange of the mechanical work of contact forces andfor dissipative fluidsof heat. III.41 Energy and entropy conservation in perfect fluids In non-dissipative non-relativistic fluids, energy is either transported convectivelyas it accompanies some flowing mass of fluidor exchanged as mechanical work of

the pressure forces between neighboring regions. Mathematically, this is expressed at the local level by the equation   ∂ 1 2 ρ(t,~r)~v(t,~r) + e(t,~r) + ρ(t,~r)Φ(t,~r) ∂t 2    1 2 ~ · +∇ ρ(t,~r)~v(t,~r) + e(t,~r) + P (t,~r) + ρ(t,~r)Φ(t,~r) ~v(t,~r) = 0, 2 (III.33) where e denotes the local density of internal energy and Φ the potential energy per unit mass of volume forcesassumed to be conservativesuch that the acceleration ~aV present in Eq. (III19) ~ equals −∇Φ. Equation (III.33) will not be proven herewe shall see later in Sec IX33 that it emerges as low-velocity limit of one of the equations of non-dissipative relativistic fluid dynamics. It is however clearly of the usual form for a conservation equation, involving • the total energy density, consisting of the kinetic ( 21 ρ~v2 ), internal (e) and potential (ρΦ) energy densities; and • the total energy flux density, which involves the previous three forms of energy, as well as that exchanged as

mechanical work of the pressure forces.(8) Remarks: ∗ The presence of pressure in the flux density, however not in the density, is reminiscent of the same property in definitions (III.21) ∗ The assumption that the volume forces are conservative is of course not innocuous. For instance, it does not hold for Coriolis forces, which means that one must be careful when working in a rotating reference frame. (8) Remember that when a system with pressure P increases its volume by an amount dV , it exerts a mechanical work P dV , “provided” to its exterior. 40 Fundamental equations of non-relativistic fluid dynamics ∗ The careful reader will have noticed that energy conservation (III.33) constitutes a fifth equation complementing the continuity and Euler equations (III.9) and (III18), yet at the cost of introducing a new scalar field, the energy density, so that now a sixth equation is needed. The latter is provided by the thermal equation of state of the fluid, which relates

its energy density, mass density and pressure.(9) In contrast to the other equations, this equation of state is not “dynamical”, ie for instance it does not involve time or spatial derivatives, but is purely algebraic. One can showagain, this will be done in the relativistic case (§ IX.32), can also be seen as special case of the result obtained for Newtonian fluids in Sec III43that in a perfect, non-dissipative fluid, the relation (III.33) expressing energy conservation locally, together with thermodynamic relations, lead to the local conservation of entropy, expressed as  ∂s(t,~r) ~  + ∇ · s(t,~r)~v(t,~r) = 0, ∂t (III.34) where s(t,~r) is the entropy density, while s(t,~r)~v(t,~r) represents the entropy flux density. The motion of a perfect fluid is thus automatically isentropic. This equation, together with a thermodynamic relation, is sometimes more practical than the energy conservation equation (III.33), to which it is however totally equivalent III.42 Energy

conservation in Newtonian fluids In a real fluid, there exist further mechanisms for transporting energy besides the convective transport due to the fluid motion, namely diffusion, either of momentum or of energy: • On the one hand, the viscous friction forces in the fluid, which already lead to the transport of momentum between neighboring fluid layers moving with different velocities, exert some work in the motion, which induces a diffusive transport of energy. This is accounted by a P i for j π contribution ·~v to the energy flux densitycomponent-wise, a contribution j π j v to the i-th component of the flux density, with π the viscous stress tensor, given in the case of a Newtonian fluid by Eq. (III26c) • On the other hand, there is also heat conduction from the regions with higher temperatures towards those with lower temperatures. This transport is described by the introduction in ~ (t,~r)in accordance with the energy flux density of a heat current(xlii) ~Q (t,~r) =

−κ(t,~r) ∇T (j) the local formulation of Fourier’s law , see e.g Sec 121 in Ref [2], with κ the heat conductivity (xliii) of the fluid. Taking into account these additional contributions, the local formulation of energy conservation in a Newtonian fluid in the absence of external volume forces reads   ∂ 1 2 ρ(t,~r)~v(t,~r) + e(t,~r) ∂t 2   1 2 ~ ρ(t,~r)~v(t,~r) + e(t,~r) + P (t,~r) ~v(t,~r) +∇ · 2    (III.35)  2 ~ v(t,~ r) ~ ~v(t,~r) + ∇ ~ − η(t,~r) ~v(t,~r) · ∇ 2      2η(t,~r) ~ ~ − ζ(t,~r) − ∇ · ~v(t,~r) ~v(t,~r) − κ(t,~r) ∇T (t,~r) = 0. 3 (9) This is where the assumption of local thermodynamic equilibrium (§ I.13) plays a crucial role (xlii) (j) Wärmestromvektor (xliii) J. B Fourier, 1768–1830 Wärmeleitfähigkeit 41 III.4 Energy conservation, entropy balance If the three transport coefficients η, ζ and κ vanish, this equation simplifies to that for perfect fluids, Eq. (III33) Remark: The energy flux density

can be read off Eq. (III35), since it represents the term between curly brackets. One can check that it can also be written as   1 2 ρ(t,~r)~v(t,~r) + e(t,~r) + P (t,~r) ~v(t,~r) − 2η(t,~r) S (t,~r) ·~v(t,~r) 2   ~ (t,~r), (III.36) ~ ·~v(t,~r) ~v(t,~r) − κ(t,~r) ∇T − ζ(t,~r) ∇ with S (t,~r) the traceless symmetric rate-of-shear tensor. One recognizes the various physical sources of energy transport. III.43 Entropy balance in Newtonian fluids In a real fluid, with viscous friction forces and heat conductivity, one can expect a priori that the transformation of mechanical energy into heat will lead in general to an increase in entropy, provided a closed system is being considered. Consider a volume V of flowing Newtonian fluid, delimited by a surface S at each point ~r of which the boundary conditions ~v(t,~r) ·~en (~r) = 0 and ~Q (t,~r) ·~en (~r) = 0 hold, where ~en (~r) denotes the unit normal vector to S at ~r. Physically, these boundary conditions mean than

neither matter nor heat flows across the surface S, so that the system inside S is closed and isolated. To completely exclude energy exchanges with the exterior of S, it is also assumed that there are no volume forces acting on the fluid inside volume V . We shall investigate the implications of the continuity equation (III10), the Navier–Stokes equation (III.31), and the energy conservation equation (III35) for the total entropy S of the fluid inside V . For the sake of brevity, the variables t, ~r will be omitted in the remainder of this Section. Starting with the energy conservation equation (III.35), the contribution      ∂ 1 2 1 2 ~ ρ~v + ∇ · ρ~v ~v ∂t 2 2 in its first two lines can be replaced by  i  3 3
i X
i 2 X
i ∂~v 1 ∂ρ 2 1 h~ ∂v ~ ~ ~v + ∇ · ρ~v ~v + ρvi ~v · ∇ v = ρvi + + ~v · ∇ v , ρ~v · ∂t 2 ∂t 2 ∂t i=1 (III.37a) i=1 where the continuity equation (III.9) was used As recalled in Appendix A, the fundamental

thermodynamic relation U = T S − PV + µN gives on the one hand e + P = T s + µn , which leads to  




~ · (e + P )~v = T ∇ ~ · s~v + µ∇ ~ · n~v +~v · s∇T ~ + n ∇µ ~ ~ · s~v + µ∇ ~ · n~v +~v · ∇ ~ P , (III.37b) ∇ = T∇ where the second identity follows from the Gibbs–Duhem relation dP = s dT + n dµ. On the other hand, it leads to de = T ds + µ dn , which under consideration of the continuity equation for particle number yields
∂e ∂s ∂n ∂s ~ · n~v . =T +µ =T − µ∇ (III.37c) ∂t ∂t ∂t ∂t With the help of relations (III.37a)–(III37c), the energy conservation equation (III35) can be rewritten as  i  3 X

i ∂s ∂v ~ · s~v +~v · ∇ ~P = ~ ρvi + ~v · ∇ v + T + T∇ ∂t ∂t i=1   X   i 3 3 X

∂ ∂v ∂ vj 2 ij ~ ∂  ~ ~ · κ∇T ~ η + − g ∇ ·~ v v + ζ ∇ ·~ v v + ∇ . (III37d) i i ∂xj ∂xj ∂xi 3 ∂xi i,j=1 i=1 42 Fundamental equations of non-relativistic fluid dynamics

Multiplying the i-th component of Eq. (III30b) by vi gives   i   i  3 X

i ∂ ∂ vj 2 ij ~ ∂ ∂v ∂P ∂v ~ ·~v . ~ vi j η = + − g ∇ ·~v + vi i ζ ∇ ρvi + ~v · ∇ v + vi ∂t ∂xi ∂x ∂xj ∂xi 3 ∂x j=1 Subtracting this identity, summed over i = 1, 2, 3, from Eq. (III37d), yields   3 X


∂ vj 2 ij ~ ∂s ∂ vi ∂ vi ~ ·~v 2 + ∇ ~ · κ∇T ~ ~ · s~v = η + − ∇ · ~ v + ζ ∇ . T + T∇ g ∂t ∂xj ∂xj ∂xi 3 (III.38) i,j=1 On the right hand side of this equation, one may use the identity    3  3  X ∂ vj 1 X ∂ vi ∂ vj 2 ij ~ ∂ vi ∂ vj 2 ~ ∂ vi ∂ vj 2 ij ~ + − g ∇·~v + − g ∇·~v + i − gij ∇·~v = , (III.39a) j 2 ∂xj ∂xi 3 ∂x ∂x 3 ∂xj ∂xi 3 ∂xi i,j=1 i,j=1 which follows from the fact that both symmetric terms ∂ vi /∂xj and ∂ vj /∂xi from the left member give the same contribution, while the term in gij yields a zero contribution, since it multiplies a traceless term.

Additionally, one has  ~ 
κ ~
2 κ∇T ~ ~ ~ + ∇T . (III.39b) ∇ · κ∇T = T ∇ · T T All in all, Eqs. (III38) and (III39) lead to    ~  3  j i X
∂ v η ∂ v 2 2 ∂s ~ ∂ v κ ∇T ∂ v j i ij ~ ·~v ~ · ~v ~ · + ∇ · s~v − ∇ = + − g ∇ + − gij ∇ ∂t T 2T ∂xj ∂xi 3 ∂xj ∂xi 3 i,j=1
~ 2
2 ∇T ζ ~ + . ∇ ·~v + κ (III.40a) T T2 This may still be recast in the slightly more compact form  ~ (t,~r)  ∇T ∂s(t,~r) ~ + ∇ · s(t,~r)~v(t,~r) − κ(t,~r) = ∂t T (t,~r) (  )  ~ (t,~r) 2 2  ∇T 1 ~ ·~v(t,~r) + κ(t,~r) 2η(t,~r) S (t,~r) : S (t,~r) + ζ(t,~r) ∇ T (t,~r) T (t,~r) (III.40b) with S : S ≡ S ij S ij the scalar obtained by doubly contracting the rate-of-shear tensor with itself. This equation can then be integrated over the V occupied by the fluid: • When computing the integral of the divergence term on the left hand side with the Stokes theorem, it vanishes thanks to the boundary conditions imposed at the

surface S; • the remaining term in the left member is simply the time derivative dS/dt of the total entropy of the closed system; • if all three transport coefficients η, ζ, κ are positive, then it is also the case of the three terms on the right hand side. One thus finds dS ≥ 0, in agreement with the second law of thermodynamics. dt Remarks: ∗ The previous derivation may be seen as a proof that the transport coefficients must be positive to ensure that the second law of thermodynamics holds. ∗ If all three transport coefficients η, ζ, κ vanish, i.e in the case of a non-dissipative fluid, Eq. (III40) simply reduces to the entropy conservation equation in perfect fluids (III34) III.4 Energy conservation, entropy balance 43 Bibliography for Chapter III • Feynman [8, 9] Chapter 40–2 & 41–1, 41–2. • Guyon et al. [2] Chapters 33, 41–43, 51, 52 • Landau–Lifshitz [3, 4] Chapter I § 1,2 & § 6,7 (perfect fluids) and Chapters II § 15,16 & V

§ 49 (Newtonian fluids). • Fließbach [13] Chapter 32. C HAPTER IV Non-relativistic flows of perfect fluids In the previous Chapter, we have introduced the coupled dynamical equations that govern the flows of perfect fluids in the non-relativistic regime, namely the continuity (III.9), Euler (III18), and energy conservation (III.33) equations for the mass density ρ(t,~r), fluid velocity ~v(t,~r) and pressure P (t,~r). The present Chapter discusses solutions of that system of equations, ie possible motions of perfect fluids,(10) obtained when using various assumptions to simplify the problem so as to render the equations tractable analytically. In the simplest possible case, there is simply no motion at all in the fluid; yet the volume forces acting at each point still drive the behavior of the pressure and local mass density throughout the medium (Sec. IV1) Steady flows, in which there is by definition no real dynamics, are also easily dealt with: both the Euler and energy

conservation equations yield the Bernoulli equation, which can be further simplified by kinematic assumptions on the flow (Sec. IV2) Section IV.3 deals with the dynamics of vortices, ie of the vorticity vector field, in the motion of a perfect fluid. In such fluids, in case the pressure only depends on the mass density, there exists a quantity, related to vorticity, that remains conserved if the volume forces at play are conservative. The latter assumption is also necessary to define potential flows (Sec. IV4), in which the further hypothesis of an incompressible motion leads to simplified equations of motion, for which a number of exact mathematical results are known, especially in the case of two-dimensional flows. Throughout the Chapter, it is assumed that the body forces in the fluid, whose volume density was denoted by f~V in Chapter III, are conservative, so that they derive from a potential. More specifically, anticipating the fact that these volume forces are proportional to

the mass they act upon, we introduce the potential energy per unit mass Φ, such that ~ f~V (t,~r) = −ρ(t,~r) ∇Φ(t,~ r). (IV.1) IV.1 Hydrostatics of a perfect fluid The simplest possibility is that of static solutions of the system of equations governing the dynamics of perfect fluids, namely those with ~v = ~0 everywherein an appropriate global reference frameand additionally ∂/∂t = 0. Accordingly, there is no strictly speaking fluid flow: this is the regime of hydrostatics, for which the only(11) non-trivial equationfollowing from the Euler equation (III.18)reads 1 ~ ~ r). ∇P (~r) = −∇Φ(~ ρ(~r) (IV.2) Throughout this Section, we adopt a fixed system of Cartesian coordinates (x1 , x2 , x3 ) = (x, y, z), with the basis vector ~e3 oriented vertically and pointing upwards. In the following examples, we shall consider the case of fluids in a homogeneous gravity field, leading to Φ(~r) = gz, with g = 9.8 m· s−2 (10) . at least in an idealized world Yet the

reader is encouraged to relate the results to observations of her everyday lifebeyond the few illustrative examples provided by the author, and to wonder how a small set of seemingly “simple” mathematical equations can describe a wide variety of physical phenomena. (11) This is true only in the case of perfect fluids; for dissipative ones, there emerge new possibilities, see Sec. VI11 45 IV.1 Hydrostatics of a perfect fluid Remark: If the stationarity condition is relaxed, the continuity equation still leads to ∂ρ/∂t = 0, i.e to a time-independent mass density Whether time derivatives vanish or not makes no change in the Euler equation when ~v = ~0. Eventually, energy conservation requires that the internal energy density eand thereby the pressurefollow the same time evolution as the “external” potential energy Φ. Thus, there is a non-stationary hydrostatics, but in which the time evolution decouples from the spatial problem. IV.11 Incompressible fluid Consider

first an incompressible fluidor, more correctly, a fluid whose compressibility can as a first approximation be neglectedwith constant, uniform mass density ρ. The fundamental equation of hydrostatics (IV.2) then yields ∂ P (~r) ∂ P (~r) = = 0, ∂x ∂y ∂ P (~r) = −ρg, ∂z i.e one recovers Pascal’s law(k) P (~r) = P (z) = P 0 − ρgz, (IV.3) with P 0 the pressure at the reference point with altitude z = 0. For instance, the reader is probably aware that at a depth of 10 m under water (ρ = 103 kg·m−3 ), the pressure is P (−10 m) = P (0) + 103 · g · 10 ≈ 2 × 105 Pa, with P (0) ≈ 105 Pa the typical atmospheric pressure at sea level. IV.12 Fluid at thermal equilibrium To depart from the assumption of incompressibility, whose range of validity is quite limited, let us instead consider a fluid at (global) thermal equilibrium i.e with a uniform temperature T ; for instance, an ideal gas, obeying the thermal equation of state PV = N kB T . Denoting by m the

mass of a molecule of that gas, the mass density is related to pressure and temperature by ρ = mP /kB T , so that Eq. (IV2) reads ∂ P (~r) ∂ P (~r) = = 0, ∂x ∂y ∂ P (~r) mg =− P (~r), ∂z kB T i.e one obtains the barotropic formula (xliv)   mgz P (~r) = P (z) = P 0 exp − . kB T Invoking the equation of state, one sees that the molecule number density n (~r) is also exponentially distributed, in agreement with the Maxwell distribution of statistical mechanics since mgz is the potential gravitational energy of a molecule at altitude z. Taking as example airwhich is a fictive ideal gas with molar mass(12) NA mair = 29 g · mol−1 the ratio kB T /mair g equals 8.8 × 103 m for T = 300 K, ie the pressure drops by a factor 2 for every elevation gain of ca. 6 km Obviously, however, assuming a constant temperature in the Earth atmosphere over such a length scale is unrealistic. (12) NA denotes the Avogadro number. (xliv) (k) barometrische Höhenformel B. Pascal,

1623–1662 46 Non-relativistic flows of perfect fluids IV.13 Isentropic fluid Let us now assume that the entropy per particle is constant throughout the perfect fluid at rest under study: s/n = constant, with s the entropy density and n the particle number density. We shall show in § IX.32 that the ratio s/n is always conserved in the motion of a relativistic perfect fluid. Taking the low-velocity limit, one deduces the conservation of s/n in a nonrelativistic non-dissipative flow: D(s/n )/Dt = 0, implying that s/n is constant along pathlines, i.e in the stationary regime along streamlines Here we assume that s/n is constant everywhere Consider now the enthalpy H = U + PV of the fluid, whose change in an infinitesimal process is the (exact) differential dH = T dS + V dP + µ dN .(13) In this relation, µ denotes the chemical potential, which will however play no further role as we assume that the number of molecules in the fluid is constant, leading to dN = 0. Dividing by N

thus gives     V S H dP , =Td + d N N N where the first term on the right-hand side vanishes since S/N is assumed to be constant. Dividing now by the mass of a molecule of the fluid, one finds   1 w d = dP , (IV.4) ρ ρ where w denotes the enthalpy density. This identity relates the change in enthalpy pro unit mass w/ρ to the change in pressure P in an elementary isentropic process. If one considers a fluid at local thermodynamic equilibrium, in which w/ρ and P takes different values at different places, the identity relates the difference in w/ρ to that in P between two (neighboring) points. Dividing by the distance between the two points, and in the limit where this distance vanishes, one derives an identity similar to (IV.4) with gradients instead of differentials. Together with Eq. (IV2), one thus obtains   ~ w(~r) + Φ(~r) = ~0 (IV.5) ∇ ρ(~r) that is w(z) + gz = constant. ρ(z) Taking as example an ideal diatomic gas, its internal energy is U = 25 N kB T ,

resulting in the enthalpy density 7 5 w = e + P = n kB T + n kB T = n kB T. 2 2 w 7 kB T That is, = , with m the mass of a molecule of gas. Equation (IV5) then gives ρ 2 m mg dT (z) =−7 . dz 2 kB In the case of air, the term on the right hand side equals 9.77 × 10−3 K · m−1 = 977 K · km−1 , i.e the temperature drops by ca 10 degrees for an elevation gain of 1 km This represents a much better modeling of the (lower) Earth atmosphere as the isothermal assumption of Sec. IV12 Remarks: ∗ The International Standard Atmosphere (ISA)(14) model of the Earth atmosphere assumes a (piecewise) linear dependence of the temperature on the altitude. The adopted value of the tem(13) (14) The reader in need of a short reminder on thermodynamics is referred to Appendix A. See e.g https://enwikipediaorg/wiki/International Standard Atmosphere 47 IV.1 Hydrostatics of a perfect fluid perature gradient in the troposphere is smaller than the above, namely 6.5 K · km−1 , to take into

account the possible condensation of water vapor into droplets or even ice. ∗ Coming back to the derivation of relation (IV.5), if we had not assumed s/n constant, we would have found     1 ~ s(~r) w(~r) ~ ~ − T (~r) ∇ , (IV.6) ∇P (~r) = ∇ ρ(~r) ρ(~r) ρ(~r) which we shall use in Sec. IV21 IV.14 Archimedes’ principle Consider now a fluid, or a system of several fluids, at rest, occupying some region of space. Let S be a closed control surface inside that region, as depicted in Fig. IV1 (left), and V be the volume delimited by S. The fluid inside S will be denoted by Σ, and that outside by Σ0 F~ S fluid 2 fluid 2 solid body G fluid 1 fluid 1 Figure IV.1 – Gedankenexperiment to illustrate Archimedes’ principle The system Σ is in mechanical equilibrium, i.e the sum of the gravity forces acting at each point of the volume V and the pressure forces exerted at each point of S by the fluid Σ0 must vanish: • The gravity forces at each point result in a

single force F~G , applied at the center of mass G of Σ, whose direction and magnitude are those of the weight of the system Σ. • According to the equilibrium condition, the resultant of the pressure forces must equal −F~G : I ~ = −F~G . P (~r) d2 S S If one now replaces the fluid system Σ by a (solid) body B, while keeping the fluids Σ0 outside S in the same equilibrium state, the mechanical stresses inside Σ0 remain unchanged. Thus, the resultant of the contact forces exerted by Σ0 on B is still given by F~ = −F~G and still applies at the center of mass G of the fluid system Σ. This constitutes the celebrated Archimedes principle: Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. (IV.7) In addition, we have obtained the point of application of the resultant force (“buoyancy” (xlv) ) from the fluid. Remark: If the center of mass G of the “removed” fluid system does not

coincide with the center of mass of the body B, the latter will be submitted to a torque, since F~ and its weight are applied at two different points. (xlv) statischer Auftrieb 48 Non-relativistic flows of perfect fluids IV.2 Steady inviscid flows We now turn to stationary solutions of the equations of motion for perfect fluids: all partial time derivatives vanishand accordingly we shall drop the t variable, yet the flow velocity~v(~r) may now be non-zero. Under those conditions, the equations (III18) and (III33) expressing the conservations of momentum and energy collapse onto a single equation (Sec. IV21) Some applications of the latter in the particular case of an incompressible fluid are then presented (Sec. IV22) IV.21 Bernoulli equation Replacing in the Euler equation (III.20) the pressure term with the help of relation (IV6) and the acceleration due to volume forces by its expression in term of the potential energy per unit mass, one finds       ∂~v(t,~r) ~

~v(t,~r)2 ~ s(t,~r) − ∇ ~ w(t,~r) − ∇Φ(t,~ ~ − ~v(t,~r) × ω ~ (t,~r) = T (t,~r) ∇ r), (IV.8) +∇ ∂t 2 ρ(t,~r) ρ(t,~r) which is rather more clumsy than the starting point (III.20), due to the many thermodynamic quantities it involves on its right hand side. Gathering all gradient terms together, one obtains     ∂~v(t,~r) ~ ~v(t,~r)2 w(t,~r) s(t,~r) ~ +∇ + + Φ(t,~r) = ~v(t,~r) × ω ~ (t,~r) + T (t,~r) ∇ . (IV.9) ∂t 2 ρ(t,~r) ρ(t,~r) In the stationary regime, the first term on the left-hand side disappears(15) and we now omit the time variable from the equations. Let d~`(~r) denote a vector tangential to the streamline at position ~r, i.e parallel to ~v(~r) When considering the scalar product of d~`(~r) with Eq. (IV9), both terms on the right hand side yield zero First, the mixed product d~`(~r) · [~v(~r) × ω ~ (~r)] is zero for it involves two collinear vectors. Second, ~ r)/ρ(~r)] vanishes due to the conservation of s/n in flows of perfect fluids,

which together d~`(~r) · ∇[s(~ ~ r)/n (~r)] = 0, where n is proportional to ρ. with the stationarity reads ~v(~r) · ∇[s(~ ~ ~ On the other hand, d`(~r) · ∇ represents the derivative along the direction of d~`, i.e along the streamline at ~r. Thus, the derivative of the term in squared brackets on the left hand side of Eq. (IV9) vanishes along a streamline, ie the term remains constant on a streamline: ~v(~r)2 w(~r) + + Φ(~r) = constant along a streamline 2 ρ(~r) (IV.10) where the value of the constant depends on the streamline. Relation (IV10) is referred as to the Bernoulli equation.(m) In the stationary regime, the energy conservation equation (III.33), in which one recognizes the enthalpy density w(~r) = e(~r) + P (~r) in the flux term, leads to the same relation (IV.10) The first term in Eq. (III33) vanishes due to the stationarity condition, leaving (we drop the variables)  2   ~v w ~ + + Φ ρ~v = 0. ∇· 2 ρ ~ · (ρ~v) Applying the product rule to the left

member, one finds a first term proportional to ∇ which vanishes thanks to the continuity equation (III.9), leaving only the other term, which is precisely ρ times the derivative along ~v of the left hand side of the Bernoulli equation.  (15) This yields a relation known as Crocco’s theorem (xlvi)(l) (xlvi) (l) Croccos Wirbelsatz L. Crocco, 1909–1986 (m) D. Bernoulli, 1700–1782 49 IV.2 Steady inviscid flows Bernoulli equation in particular cases ::::::::::::::::::::::::::::::::::::::: Coming back to Eq. (IV9), if the steady flow is irrotational, ie ω ~ (~r) = ~0 everywhere, and isentropic, i.e s(~r)/n (~r) is uniform, then the gradient on the left hand side vanishes That is, the constant in the Bernoulli equation (IV.10) is independent of the streamline, ie it is the same everywhere. ~ ·~v(~r) = 0, then the continuity equation shows that the In case the flow is incompressible, i.e ∇ mass density ρ becomes uniform throughout the fluid. One may then replace

pull the factor 1/ρ inside the pressure gradient in the Euler equation (III.20) Repeating then the same steps as below Eq. (IV9), one finds that the Bernoulli equation now reads In incompressible flows ~v(~r)2 P (~r) + + Φ(~r) is constant along a streamline. 2 ρ (IV.11) This is the form which we shall use in the applications hereafter. Can this form be reconciled with the other one (IV.10), which is still what follows from the energy conservation equation? Subtracting one from the other, one finds that the ratio e(~r)/ρ is constant along streamlines. That is, since ρ is uniform, the internal energy density is constant along pathlineswhich coincide with streamlines in a steady flow Now, thermodynamics expresses the differential de through ds and dn : since both entropy and particle number are conserved along a pathline, so is internal energy, i.e Eq (IV10) is compatible with Eq (IV11) IV.22 Applications of the Bernoulli equation Throughout this Section, we assume that the flow

is incompressible, i.e the mass density is uniform, and rely on Eq. (IV11) Of course, one may release this assumption, in which case one should replace pressure by enthalpy density everywhere below.(16) IV.22 a Drainage of a vessel. Torricelli’s law :::::::::::::::::::::::::::::::::::::::::::: Consider a liquid contained in a vessel with a small hole at its bottom, through which the liquid can flow (Fig. IV2) A 6 h B ? Figure IV.2 At points A and B, which lie on the same streamline, the pressure in the liquid equals the atmospheric pressure(17) P A = P B = P 0 . The Bernoulli equation (at constant pressure) then yields vA2 v2 + gzA = B + gzB , 2 2 with zA resp. zB the height of point A resp B, ie vB2 = vA2 + 2gh. If the velocity at point A vanishes, one finds Torricelli’s law (xlvii)(n) p vB = 2gh. That is, the speed of efflux is the same as that acquired by a body in free fall from the same height h in the same gravity field. (16) (17) The author confesses that he has a

better physical intuition of pressure than of enthalpy, hence his parti pris. One can show that the pressure in the liquid at point B equals the atmospheric pressure provided the local streamlines are parallel to each otherthat is, the flow is laminar. (xlvii) (n) Torricellis Theorem E. Torricelli, 1608–1647 50 Non-relativistic flows of perfect fluids Remark: To be allowed to apply the Bernoulli equation, one should first show that the liquid flows steadily. If the horizontal cross section of the vessel is much larger than the aperture of the hole and h large enough, this holds to a good approximation. IV.22 b Venturi effect ::::::::::::::::::::: Consider now the incompressible flow of a fluid in the geometry illustrated in Fig. IV3 As we shall only be interested in the average velocity or pressure of the fluid across a cross section of the tube, the flow is effectively one-dimensional. @ @ v1- s S v2 - Figure IV.3 The conservation of the mass flow rate in the tube,

which represents the integral formulation of the continuity equation (III.9), leads to ρSv1 = ρ s v2 , ie v2 = (S/s )v1 > v1 , with S resp s the area of the tube cross section in its broad resp. narrow section On the other hand, the Bernoulli equation at constant height, and thus potential energy, gives v2 P 2 v12 P 1 + = 2+ . 2 ρ 2 ρ All in all, the pressure in the narrow section is thus smaller than in the broad section, P 2 < P 1 , which constitutes the Venturi effect.(o) Using mass conservation and the Bernoulli equation, one can express v1 or v2 in terms of the tube cross section areas and the pressure difference. For instance, the mass flow rate reads   1/2 P 1 −P 2 S 2 ρS 2 . − 1 ρ s2 IV.22 c Pitot tube :::::::::::::::::: Figure IV.4 represents schematically the flow of a fluid around a Pitot tube,(p) which is a device used to estimate a flow velocity through the measurement of a pressure difference. Three streamlines are shown, starting far away from the

Pitot tube, where the flow is (approximately) uniform and has the velocity ~v, which one wants to measure. The flow is assumed to be incompressible - manometer ( O• 0 O• - I• 0 A • • A - • B - ~v Figure IV.4 – Flow around a Pitot tube The Pitot tube consists of two long thin concentric tubes: • Despite the presence of the hole at the end point I, the flow does not penetrate in the inner tube, so that ~vI = ~0 to a good approximation. (o) G. B Venturi, 1746–1822 (p) H. Pitot, 1695–1771 51 IV.2 Steady inviscid flows • In the broader tube, there is a hole at a point A, which is far enough from I to ensure that the fluid flow in the vicinity of A is no longer perturbed by the extremity of the tube: ~vA =~vA0 ~v, where the second identity holds thanks to the thinness of the tube, which thus perturbs the flow properties minimally. In addition, the pressure in the broader tube is uniform, so that P = PB . If one neglects the height differenceswhich

is a posteriori justified by the numerical values we shall findthe (incompressible) Bernoulli equation gives first PO + ρ along the streamline OI, and ~v2 = PI 2 ~v2 0 ~v2 = PA0 + ρ A 2 2 PO , PA0 PA and ~vA0 ~v, the latter identity leads to PO0 + ρ along the streamline O0 A0 . Using PO0 PO PA = PB . One thus finds ~v2 , 2 so that a measurement of PI − PB gives an estimate of |~v|. PI − PB = ρ For instance, in air (ρ ∼ 1.3 kg · m−3 ) a velocity of 100 m · s−1 results in a pressure difference of 6.5 × 103 Pa = 65 × 10−2 atm With a height difference h of a few centimeters between O and A0 , the neglected term ρgh is of order 1 Pa. Remarks: ∗ The flow of a fluid with velocity ~v around a motionless Pitot tube is equivalent to the motion of a Pitot tube with velocity −~v in a fluid at rest. Thus Pitot tubes are used to measure the speed of airplanes. ∗ Is the flow of air really incompressible at velocities of 100 m · s−1 or higher? Not really,

since the Mach number (II.16) becomes larger than 03 In practice, one thus rather uses the “compressible” Bernoulli equation (IV.10), yet the basic principles presented above remain valid IV.22 d Magnus effect Consider an initially uniform and steady flow with velocity ~v0 . One introduces in it a cylinder, which rotates about its axis with angular velocity ω ~ C perpendicular to the flow velocity (Fig. IV5) :::::::::::::::::::::: ~v0 ω ~C Figure IV.5 – Fluid flow around a rotating cylinder Intuitively, one can expect that the cylinder will drag the neighboring fluid layers along in its rotation.(18) In that case, the fluid velocity due to that rotation will add up to resp be subtracted from the initial flow velocity in the lower resp. upper region close to the cylinder in Fig IV5 (18) Strictly speaking, this is not true in perfect fluids, only in real fluids with friction! Nevertheless, the tangential forces due to viscosity in the latter may be small enough that the

Bernoulli equation remains approximately valid, as is assumed here. 52 Non-relativistic flows of perfect fluids Invoking now the Bernoulli equationin which the height difference between both sides of the cylinder is neglected, the pressure will be larger above the cylinder than below it. Accordingly, the cylinder will experience a resulting force directed downwardsmore precisely, it is proportional to ~v0 × ω ~ C , which constitutes the Magnus effect.(q) IV.3 Vortex dynamics in perfect fluids We now turn back to the case of an arbitrary flow ~v(t,~r), still in the case of a perfect fluid. The vorticity vector field, defined as the rotational curl of the flow velocity field, was introduced in Sec. II12, together with the vorticity lines Modulo a few assumptions on the fluid equation of state and the volume forces, one can show that vorticity is “frozen” in the flow of a perfect fluid, in the sense that there the flux of vorticity across a material surface remains constant

as the latter is being transported. This behavior will be investigated and formulated both at the integral level (Sec. IV31) and differentially (Sec IV32) IV.31 Circulation of the flow velocity Kelvin’s theorem Definition: Let ~ γ (t, λ) be a closed curve, parametrized by a real number λ ∈ [0, 1], which is being swept along by the fluid in its motion. The integral I Γ~γ (t) ≡ ~v(t, ~γ (t, λ)) · d~` (IV.12) ~γ is called the circulation around the curve of the velocity field. Remark: According to Stokes’ theorem,(19) if the area bounded by the contour ~ γ (t, λ) is simply connected, Γ~γ (t) equals the surface integralthe “flux”of the vorticity field over every surface S~γ (t) delimited by ~γ : Z Z  2  ~ ~ ~ Γ~γ (t) = (IV.13) ω ~ (t,~r) · d2 S. ∇ ×~v(t,~r) · d S = S~γ S~γ Stated differently, the vorticity field is the flux density of the circulation of the velocity. This relationship between circulation and vorticity will be further exploited

hereafter: we shall now establish and formulate results at the integral level, namely for the circulation, which will then be expressed at the differential level, i.e in terms of the vorticity, in Sec IV32 Many results take a simpler form in a so-called barotropic fluid ,(xlviii) in which the pressure can be expressed as function of only the mass density: P = P (ρ), irrespective of whether the fluid is otherwise perfect or dissipative. An example of such a result is Kelvin’s circulation theorem:(r) In a perfect barotropic fluid with conservative volume forces, the circulation of (IV.14a) the flow velocity around a closed curve comoving with the fluid is conserved. Denoting by ~γ (t, λ) the closed contour in the theorem, DΓ~γ (t) = 0. Dt This result is also sometimes called Thomson’s theorem. (19) which in its classical form used here is also known as Kelvin–Stokes theorem. (xlviii) (q) barotropes Fluid G. Magnus, 1802–1870 (r) W. Thomson, Baron Kelvin, 1824–1907

(IV.14b) 53 IV.3 Vortex dynamics in perfect fluids Proof: For the sake of brevity, the arguments of the fields are omitted in case it is not necessary to specify them. Differentiating definition (IV12) first gives   Z Z 1 2 DΓ~γ D 1 ∂~γ (t, λ) ∂~γ ∂~v X ∂~v ∂γ i ∂ ~γ dλ. = ·~v(t, ~γ (t, λ)) dλ = ·~v + · + Dt Dt 0 ∂λ ∂λ ∂t ∂λ ∂t ∂xi ∂t 0 i ∂~γ (t, λ) Since the contour ~γ (t, λ) flows with the fluid, =~v(t, ~γ (t, λ)), which leads to ∂t   Z 1
DΓ~γ ∂~γ ∂~v ∂~v ~ ~v dλ. = · ~v + · + ~v · ∇ Dt ∂λ ∂λ ∂t 0 The first term in the curly brackets is clearly the derivative with respect to λ of ~v2/2, so that its integral along a closed curve vanishes. The second term involves the material derivative of ~ leads to ~v, as given by the Euler equation. Using Eq (III19) with ~aV = −∇Φ  Z 1 ~ DΓ~γ ∂~γ ∇P ~ = − ∇Φ · dλ. − Dt ρ ∂λ 0 ~ around a closed contour vanishes, leaving Again, the

circulation of the gradient ∇Φ I ~ DΓ~γ (t) ∇P (t,~r) ~ =− · d`, Dt r) ~ γ ρ(t,~ (IV.15) which constitutes the general case of Kelvin’s circulation theorem for perfect fluids with conservative volume forces. Transforming the contour integral with Stokes’ theorem yields the surface integral of ~  ~ ~ P × ∇ρ ~ ~P ~ P × ∇ρ ~ ∇ ∇×∇ ∇ ∇P ~ + = = . (IV.16) ∇× ρ ρ ρ2 ρ2 ~ P and ∇ρ ~ are collinear, In a barotropic fluid, the rightmost term of this identity vanishes since ∇ which yields relation (IV.14)  Remark: Using relation (IV.13) and the fact that the area S~γ (t) bounded by the curve ~ γ at time t defines a material surface, which will be transported in the fluid motion, Kelvin’s theorem (IV.14) can be restated as In a perfect barotropic fluid with conservative volume forces, the flux of the vorticity across a material surface is conserved. (IV.17) Kelvin’s theorem leads to two trivial corollaries, namely Helmholtz’s

theorem:(s) In the flow of a perfect barotropic fluid with conservative volume forces, the vorticity lines and vorticity tubes move with the fluid. (IV.18) Similar to the definition of stream tubes in Sec. I33, a vorticity tube is defined as the surface formed by the vorticity lines tangent to a given closed geometrical curve. And in the case of vanishing vorticity ω ~ = ~0, one has Lagrange’s theorem: In a perfect barotropic fluid with conservative volume forces, if the flow is irrotational at a given instant t, it remains irrotational at later times. (IV.19) Kelvin’s circulation theorem (IV.14) and its corollaries imply that vorticity cannot be created nor destroyed in the flow of a perfect barotropic fluid with conservative volume forces. However, (s) H. von Helmholtz, 1821–1894 54 Non-relativistic flows of perfect fluids the more general form (IV.15) already show that in a non-barotropic fluid, there is a “source” for vorticity, which leads to the

non-conservation of the circulation of the flow velocity. Similarly, nonconservative forcesfor instance a Coriolis force in a rotating reference framemay contribute a non-vanishing term in Eq. (IV15) leading to a change in the circulation We shall see in Sec VI5 that viscous stresses also affect the transport of vorticity in a fluid. IV.32 Vorticity transport equation in perfect fluids ~ Consider the Euler equation (III.20), in the case of conservative volume forces, ~aV = −∇Φ Taking the rotational curl of both sides yields after some straightforward algebra ~ P (t,~r) × ∇ρ(t,~ ~   ∂~ ω (t,~r) ~ ∇ r) ~ (t,~r) = − − ∇ × ~v(t,~r) × ω . (IV.20) 2 ∂t ρ(t,~r) This relation can be further transformed using the identity (we omit the variables)




~ ω ~ ·~v ω ~ ·ω ~ ~v + ∇ ~ × ~v × ω ~− ∇ ~. ~ ~v − ~v · ∇ ~ = ω ~ ·∇ ∇ ~ ·ω Since the divergence of the vorticity field ∇ ~ (t,~r) vanishes, the previous two equations yield ~ P (t,~r)

× ∇ρ(t,~ ~      ∂~ ω (t,~r)  ∇ r) ~ ·~v(t,~r) ω ~ ω ~ ~v(t,~r) = − ∇ ~ (t,~r) − ~ (t,~r) − ω ~ (t,~r) · ∇ + ~v(t,~r) · ∇ . ∂t ρ(t,~r)2 (IV.21) While it is tempting to introduce the material derivative D~ ω /Dt on the left hand side of this equation, for the first two terms, we rather define the whole left member to be a new derivative    ∂~ ω (t,~r)  ~ (t,~r) D~v ω ~ ω ~ ~v(t,~r) ~ (t,~r) − ω ~ (t,~r) · ∇ ≡ + ~v(t,~r) · ∇ Dt ∂t (IV.22a) or equivalently  D~ ω (t,~r)  ~ (t,~r) D~v ω ~ ~v(t,~r). ≡ − ω ~ (t,~r) · ∇ (IV.22b) Dt Dt We shall refer to D~v /Dt as the comoving time derivative, for reasons that will be explained at the end of this Section. Using this definition, Eq. (IV21) reads ~ P (t,~r) × ∇ρ(t,~ ~   ∇ r) ~ (t,~r) D~v ω ~ ·~v(t,~r) ω ~ (t,~r) − =− ∇ . 2 Dt ρ(t,~r) (IV.23) In the particular of a barotropic fluidrecall that we also assumed that it is ideal and only has conservative volume

forcesthis becomes   D~v ω ~ (t,~r) ~ ·~v(t,~r) ω =− ∇ ~ (t,~r). Dt (IV.24) Thus, the comoving time-derivative of the vorticity is parallel to itself. From Eq. (IV24), one deduces at once that if ω ~ (t,~r) vanishes at some time t, it remains zero which is the differential formulation of corollary (IV.19) ~ ·~v on the right hand side Invoking the continuity equation (III.9), the volume expansion rate ∇ of Eq. (IV24) can be replaced by −(1/ρ)Dρ/Dt For scalar fields, material derivative and comoving time-derivative coincide, which leads to the compact form   D~v ω ~ (t,~r) = ~0 (IV.25) Dt ρ(t,~r) for perfect barotropic fluids with conservative volume forces. That is, anticipating on the discussion 55 IV.3 Vortex dynamics in perfect fluids of the comoving time derivative hereafter, ω ~ /ρ evolves in the fluid flow in the same way as the separation between two material neighboring points: the ratio is “frozen” in the fluid evolution. Comoving time

derivative :::::::::::::::::::::::::: To understand the meaning of the comoving time derivative D~v /Dt, let us come back to Fig. II1 depicting the positions at successive times t and t + δt of a small material vector δ~`(t). By definition of the material derivative, the change in δ~` between these two instantsas given by the trajectories of the two material points which are at ~r resp. ~r + δ~`(t) at time tis Dδ~`(t) δ~`(t+δt) − δ~`(t) = δt. Dt On the other hand, displacing the origin of δ~`(t+δt) to let it coincide with that of δ~`(t), one sees x2   ~ ~v(t,~r)δt δ~`(t)· ∇ + t, ~r ~v δ~`(t + δt) δ~`(t) ~r
δt ) t ( ~ δ` ~v(t,~r) δ t x1 x3 Figure IV.6 – Positions of a material line element δ~` at successive times t and t + δt on Fig. IV6 that this change equals   ~ ~v(t,~r)δt. δ~`(t+δt) − δ~`(t) = δ~`(t)· ∇ Equating both results and dividing by δt, one finds  Dδ~`(t)  ~ ~ ~v(t,~r), i.e precisely = δ `(t)· ∇ Dt D~v δ~`(t)

~ = 0. Dt (IV.26) Thus, the comoving time derivative of a material vector, which moves with the fluid, vanishes. In turn, the comoving time derivative at a given instant t of an arbitrary vector measures its rate of change with respect to a material vector with which it coincides at time t. This interpretation suggeststhis can be proven more rigorouslywhat the action of the comoving time derivative on a scalar field should be. In that case, D~v /Dt should coincide with the material derivative, which already accounts for all changesdue to non-stationarity and convective transportaffecting material points in their motion. This justifies a posteriori our using D~v ρ/Dt = Dρ/Dt above. More generally, the comoving time derivative introduced in Eq. (IV22a) may be rewritten as ∂ D~v ( · ) ≡ ( · ) + L~v ( · ), (IV.27) Dt ∂t where L~v denotes the Lie derivative along the velocity field ~v(~r), whose action on an arbitrary 56 Non-relativistic flows of perfect fluids vector

field ω ~ (~r) is precisely (time plays no role here)     ~ ω ~ ~v(~r), ~ (~r) ≡ ~v(~r) · ∇ ~ (~r) − ω ~ (~r) · ∇ L~v ω while it operates on an arbitrary scalar field ρ(~r) according to   ~ ρ(~r). L~v ρ(~r) ≡ ~v(~r) · ∇ More
information on the Lie derivative, including its operation on 1-forms or more generally on m n -tensorsfrom which the action of the comoving time derivative follows, can be found e.g in Ref [14, Chap 31–35] IV.4 Potential flows According to Lagrange’s theorem (IV.19), every flow of a perfect barotropic fluid with conservative volume forces which is everywhere irrotational at a given instant remains irrotational at every time. Focusing accordingly on the incompressible and irrotational motion of an ideal fluid with conservative volume forces, which is also referred to as a potential flow (xlix) , the dynamical equations can be recast such that the main one is a linear partial differential equation for the velocity potential (Sec.

IV41), for which there exist mathematical results (Sec IV42) Two-dimensional potential flows are especially interesting, since one may then introduce a complex velocity potentialand the corresponding complex velocity, which is a holomorphic function (Sec. IV43) This allows one to use the full power of complex analysis so as to devise flows around obstacles with various geometries by combining “elementary” solutions and deforming them. IV.41 Equations of motion in potential flows Using a known result from vector analysis, a vector field whose curl vanishes everywhere on a simply connected domain of R3 can be written as the gradient of a scalar field. Thus, in the case ~ ×~v(t,~r) = ~0, the velocity field can be expressed as of an irrotational flow ∇ ~ ~v(t,~r) = −∇ϕ(t,~ r) (IV.28) with ϕ(t,~r) the so-called velocity potential .(l) Remarks: ∗ The minus sign in definition (IV.28) is purely conventional While the choice adopted here is not universal, it has the

advantage of being directly analogous to the convention in electrostatics ~ = −∇Φ ~ Coul. ) or Newtonian gravitation physics (~g = −∇Φ ~ Newt. ) (E ∗ Since Lagrange’s theorem does not hold in a dissipative fluid, in which vorticity can be created or annihilated (Sec. VI5), the rationale behind the definition of the velocity potential disappears ~ expressing that the volume Using the velocity potential (IV.28) and the relation ~aV = −∇Φ forces are conservative, the Euler equation (III.20) reads ( ) 2 ~ ~ r) ∂ ∇ϕ(t,~ r) ~ ∇ϕ(t,~ 1 ~ − +∇ + Φ(t,~r) = − ∇P (t,~r). ∂t 2 ρ(t,~r) Assuming that the flow is also incompressible, and thus ρ constant, this becomes ( ) 2 ~ ~ r) P (t,~r) ∂ ∇ϕ(t,~ r) ~ ∇ϕ(t,~ +∇ + + Φ(t,~r) = ~0. − ∂t 2 ρ (xlix) Potentialströmung (l) Geschwindigkeitspotential (IV.29) 57 IV.4 Potential flows or equivalently  2 ~ ∇ϕ(t,~ r) ∂ϕ(t,~r) P (t,~r) − + + + Φ(t,~r) = C(t), ∂t 2 ρ (IV.30)

where C(t) denotes a function of time only. ~ · ~v(t,~r) = 0 leads to the Eventually, expressing the incompressibility condition [cf. Eq (II13)] ∇ (t) Laplace equation 4ϕ(t,~r) = 0 (IV.31) for the velocity potential ϕ. Equations (IV.28), (IV30) and (IV31) are the three equations of motion governing potential flows. Since the Laplace equation is partial differential, it is still necessary to specify the corresponding boundary conditions In agreement with the discussion in § III.32 c, there are two types of condition: at walls or obstacles, which are impermeable to the fluid; and “at infinity”for a flow in an unbounded domain of space, where the fluid flow is generally assumed to be uniform. Choosing a proper reference frame R, this uniform motion of the fluid may be turned into having a fluid at rest. Denoting by S(t) the material surface associated with walls or obstacles, which are assumed to be moving with velocity ~vobs. in R, and by ~en (t,~r) the unit normal vector to

S(t) at a given point ~r, the condition of vanishing relative normal velocity reads ~ − ~en (t,~r) · ∇ϕ(t,~ r) = ~en (t,~r) · ~vobs. (t,~r) on S(t) (IV.32a) In turn, the condition of rest at infinity reads ϕ(t,~r) ∼ K(t), |~r|∞ (IV.32b) where the scalar function K(t) will in practice be given. Remarks: ∗ Since the Laplace equation (IV.31) is linearthe non-linearity of the Euler equation is in Eq. (IV30), which is “trivial” once the spatial dependence of the velocity potential has been determined, it will be possible to superpose the solutions of “simple” problems to obtain the solution for a more complicated geometry. ∗ In potential flows, the dependences on time and space are somewhat separated: The Laplace equation (IV.31) governs the spatial dependence of ϕ and thus~v; meanwhile, time enters the boundary conditions (IV32), thus is used to “normalize” the solution of the Laplace equation In turn, when ϕ is known, relation (IV.30) gives the pressure

field, where the integration “constant” C(t) will also be fixed by boundary conditions. IV.42 Mathematical results on potential flows The boundary value problem consisting of the Laplace differential equation (IV.31) together with the boundary conditions on normal derivatives (IV.32) is called a Neumann problem (u) or boundary value problem of the second kind. For such problems, results on the existence and unicity of solutions have been established, which we shall now state without further proof.(20) (20) (t) The Laplace differential equation is dealt with in many textbooks, as e.g in Ref [15, Chapters 7–9], [16, Chapter 4], or [17, Chapter VII]. P.-S (de) Laplace, 1749–1827 (u) C. Neumann, 1832–1925 58 Non-relativistic flows of perfect fluids IV.42 a Potential flows in simply connected regions :::::::::::::::::::::::::::::::::::::::::::::::::::: The simplest case is that of a potential flow on a simply connected domain D of space. D may be unbounded, provided

the condition at infinity is that the fluid be at rest, Eq. (IV32b) On a simply connected domain, the Neumann problem (IV.31)–(IV32) for the velocity potential admits a solution ϕ(t,~r), which is unique up to an additive constant. (IV.33) In turn, the flow velocity field ~v(t,~r) given by relation (IV.28) is unique For a flow on a simply connected region, the relation (IV.28) between the flow velocity and its potential is “easily” invertible: fixing some reference position ~r0 in the domain, one may write Z ϕ(t,~r) = ϕ(t,~r0 ) − ~v(t,~r0 ) · d~`(~r0 ) (IV.34) ~γ where the line integral is taken along any path ~γ on D connecting the positions ~r0 and ~r. That the line integral only depends on the path extremities ~r0 , ~r, not on the path itself, is clearly equivalent to Stokes’ theorem stating that the circulation of velocity along any closed contour in the domain D is zeroit equals the flux of the vorticity, which is everywhere zero, through a surface delimited by

the contour and entirely contained in D. Thus, ϕ(t,~r) is uniquely defined once the value ϕ(t,~r0 ), which is the arbitrary additive constant mentioned above, has been fixed. This reasoning no longer holds in a multiply connected domain, as we now further discuss. IV.42 b Potential flows in doubly connected regions ::::::::::::::::::::::::::::::::::::::::::::::::::::: As a matter of fact, in a doubly (or a fortiori multiply) connected domain, there are by definition non-contractible closed paths. Considering for instance the domain D traversed by an infinite cylinderwhich is not part of the domainof Fig. IV7, the path going from ~r0 to ~r2 along ~γ02 0 (21) cannot be continuously shrunk to a point without leaving D. then coming back to ~r0 along ~γ02 This opens the possibility that the line integral in relation (IV.34) depend on the path connecting two points. ~r2 • 0 ~γ 602 00 ~γ02 6 ~γ02 6 ~γ 0 -01 •~ r1 • ~r0 - ~γ01 Figure IV.7 In a doubly connected

domain D, there is only a single “hole” that prevents closed paths from being homotopic to a point, i.e contractible Let Γ(t) denote the circulation at time t of the velocity around a closed contour, with a given “positive” orientation, circling the hole once. One easily checkse.g invoking Stokes’ theoremthat this circulation has the same value for all closed (21) 0 More precisely, if ~γ02 is parameterized by λ ∈ [0, 1] when going from ~r0 to ~r2 , a path from ~r2 to ~r0 with the 0 0 same geometric supportwhich is what is meant by “coming back along ~γ02 ”is λ 7 ~γ02 (1 − λ). IV.4 Potential flows 59 paths with the same orientation going only once around the hole, since they can be continuously deformed into each other without leaving D. Accordingly, the “universal” circulation Γ(t) is also referred to as cyclic constant (li) of the flow. More generally, the circulation at time t of the velocity around a closed curve circling the hole n times and

oriented in the positive resp. negative direction is nΓ(t) resp −nΓ(t) Going back to the line integral in Eq. IV34, its value will generally depend on the path ~γ from ~r0 to ~ror more precisely, on the class, defined by the number of loops around the hole, of the path. Illustrating this idea on Fig IV7, while the line integral from ~r0 to ~r2 along the path ~γ02 0 will have a given value I, the line integral along ~γ02 will differ by one (say, positive) unit of Γ(t) 00 and be equal to I +Γ(t). In turn, the integral along ~γ02 , which makes one more negatively oriented loop than ~γ02 around the cylinder, takes the value I − Γ(t). These preliminary discussions suggest that if the Neumann problem (IV.31)–(IV32) for the velocity potential on a doubly connected domain admits a solution ϕ(t,~r), the latter will not be a scalar function in the usual sense, but rather a multivalued function, whose various values at a given position ~r at a fixed time t differ by an integer

factor of the cyclic constant Γ(t). All in all, the following result holds provided the cyclic constant Γ(t) is known, i.e if its value at time t is part of the boundary conditions: On a doubly connected domain, the Neumann problem (IV.31)–(IV32) for the velocity potential with given cyclic constant Γ(t) admits a solution ϕ(t,~r), which is unique (IV.35) up to an additive constant. The associated flow velocity field ~v(t,~r) is unique The above wording does not specify the nature of the solution ϕ(t,~r): • if Γ(t) = 0, in which case the flow is said to be acyclic, the velocity potential ϕ(t,~r) is a univalued function; • if Γ(t) 6= 0, i.e in a cyclic flow , the velocity potential ϕ(t,~r) is a multivalued function of its spatial argument. Yet as the difference between the various values at a given ~r is function of time only, the velocity field (IV.28) remains uniquely defined Remarks: ∗ Inspecting Eq. (IV30), one might fear that the pressure field P (t,~r) be

multivalued, reflecting the term ∂ϕ(t,~r)/∂t. Actually, however, Eq (IV30) is a first integral of Eq (IV29), in which the ~r-independent multiples of Γ(t) distinguishing the multiple values of ϕ(t,~r) disappear when the gradient is taken. That is, the term ∂ϕ(t,~r)/∂t is to be taken with a grain of salt, since in fact it does not contain Γ(t) or its time derivative. ∗ In agreement with the first remark, the reader should remember that the velocity potential ϕ(t,~r) is just a useful auxiliary mathematical function,(22) yet the physical quantity is the velocity itself. Thus the possible multivaluedness of ϕ(t,~r) is not a real physical problem. (22) (li) ~ . Like its cousins: gravitational potential ΦNewt. , electrostatic potential ΦCoul , magnetic vector potential A zyklische Konstante 60 Non-relativistic flows of perfect fluids IV.43 Two-dimensional potential flows We now focus on two-dimensional potential flows, for which the velocity fieldand all other

fieldsonly depend on two coordinates. The latter will either be Cartesian coordinates (x, y), which are naturally combined into a complex variable z = x + iy, or polar coordinates (r, θ). Throughout this Section, the time variable t will not be denoted: apart from possibly influencing the boundary conditions, it plays no direct role in the determination of the velocity potential. IV.43 a Complex flow potential and complex flow velocity :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Let us first introduce a few useful auxiliary functions, which either simplify the description of two-dimensional potential flows, or allow one to “generate” such flows at will. Stream function Irrespective of whether the motion is irrotational or not, in an incompressible two-dimensional flow one can define a unique (up to an additive constant) stream function (lii) ψ(x, y) such that vx (x, y) = − ∂ψ(x, y) , ∂y vy (x, y) = ∂ψ(x, y) ∂x (IV.36) at every point (x, y).

Indeed, when the above two relations hold, the incompressibility criterion ~ ·~v(x, y) = 0 is fulfilled automatically. ∇ Remark: As in the case of the relation between the flow velocity field and the corresponding potential, Eq. (IV28), the overall sign in the relation between ~v(~r) and ψ(~r) is conventional Yet if one wishes to define the complex flow potential as in Eq. (IV39) below, the relative sign of ϕ(~r) and ψ(~r) is fixed. The stream function for a given planar fluid motion is such that the lines along which ψ(~r) is constant are precisely the streamlines of the flow. Let d~x(λ) denote a differential line element of a curve ~x(λ) of constant ψ(~r), i.e a curve along
~ = ~0. Then d~x(λ) · ∇ψ ~ ~x(λ) = 0 at every point on the line: using relations (IV.36), which ∇ψ one recovers Eq. (I15b) characterizing a streamline  Stream functions are also defined in three-dimensional flows, yet in that case two of them are needed. More precisely, one can find two

linearly independent functions ψ1 (~r), ψ2 (~r), such that the streamlines are the intersections of the surfaces of constant ψ1 and of constant ψ2 . ~ 1 (~r) × ∇ψ ~ 2 (~r), with an a priori That is, they are such that the flow velocity obeys ~v(~r) ∝ ∇ψ position-dependent proportionality factorwhich can be taken identically equal to unity in an incompressible flow. Consider now a potential flow, i.e which is not only incompressible, but also irrotational For such a two-dimensional flow, the condition of vanishing vorticity reads ∂ vy (x, y) ∂ vx (x, y) ω z (x, y) = − = 0, ∂x ∂y which under consideration of relations (IV.36) gives 4ψ(x, y) = 0 (IV.37a) at every point (x, y). That is, the stream function obeys the Laplace equationjust like the velocity potential ϕ(~r). A difference with ϕ(~r) arises with respect to the boundary conditions. At an obstacle or walls, modeled by a “surface” Sin the plane R2 , this surface is rather a curve, the

impermeability condition implies that the velocity is tangential to S, i.e S coincides with a streamline: ψ(x, y) = constant on S (lii) Stromfunktion (IV.37b) 61 IV.4 Potential flows For a flow on an unbounded domain, the velocity is assumed to be uniform at infinity, ~v(x, y) ~v∞ which is the case if y x ψ(x, y) ∼ v∞ x − v∞ y (IV.37c) |~r|∞ x , vy the components of ~v . with v∞ ∞ ∞ The boundary conditions (IV.37b)–(IV37c) on the stream function are thus dissimilar from the corresponding conditions (IV.32a)–(IV32b) on the velocity potential In particular, the condition at an obstacle involves the stream function itself, instead of its derivative: the Laplace differential equation (IV.37a) with conditions (IV37b)–(IV37c) represents a Dirichlet problem,(v) or boundary value problem of the first kind, instead of a Neumann problem. Complex flow potential In the case of a two-dimensional potential flow, both the velocity potential φ(x, y) and the stream

function ψ(x, y) are so-called harmonic functions, i.e they are solutions to the Laplace differential equation, see Eqs. (IV31) and (IV37a) In addition, gathering Eqs (IV28) and (IV36), one sees that they satisfy at every point (x, y) the identities  ∂ψ(x, y)  ∂φ(x, y) = = −vx (x, y) , ∂x ∂y  ∂φ(x, y) ∂ψ(x, y)  =− = −vy (x, y) . ∂y ∂x (IV.38) The relations between the partial derivatives of φ and ψ are precisely the Cauchy–Riemann equations obeyed by the corresponding derivatives of the real and imaginary parts of a holomorphic function of a complex variable z = x + iy. That is, the identities (IV38) suggest the introduction of a complex (flow ) potential φ(z) ≡ ϕ(x, y) + iψ(x, y) with z = x + iy (IV.39) which will automatically be holomorphic on the domain where the flow is defined. The functions ϕ and ψ are then said to be conjugate to each other. In line with that notion, the curves in the plane along which one of the functions is

constant are the field lines of the other, and reciprocally. Besides the complex potential φ(z), one also defines the corresponding complex velocity as the negative of its derivative, namely w(z) ≡ − dφ(z) = vx (x, y) − ivy (x, y) dz (IV.40) where the second identity follows at once from the definition of φ and the relations between ϕ or ψ and the flow velocity. Like φ(z), the complex velocity w(z) is an analytic function of z IV.43 b Elementary two-dimensional potential flows ::::::::::::::::::::::::::::::::::::::::::::::::::::: As a converse to the above construction of the complex potential, the real and imaginary parts of any analytic function of a complex variable are harmonic functions, i.e any analytical function φ(z) defines a two-dimensional potential flow on its domain of definition. Accordingly, we now investigate a few “basic” complex potentials and the flows they describe. Uniform flow The simplest possibility is that of a linear complex potential:

φ(z) = −v e−iα z (v) P. G (Lejeune-)Dirichlet, 1805–1859 with v ∈ R, α ∈ R. (IV.41) 62 Non-relativistic flows of perfect fluids y   6             3  3  3  3  3 3 3               α          3  3  3  3  3  3  3           x        3  3 3 3 3 3 3 3                     3  3  3  3  3  3   Using for instance Eq. (IV40), this trivially leads to a uniform velocity field making an angle α with the x-direction,
~v(x, y) = cos α~ex + sin α~ey v, as illustrated in Fig. IV8, in which a few streamlines are displayed, to which the equipotential lines (not shown) of ϕ(x, y) are perpendicular. Figure IV.8 Flow source or sink Another flow with “simple” streamlines is that defined by the complex potential(23) Q log(z − z0 ) with Q ∈ R, z0 ∈ C. 2π The resulting complex flow velocity φ(z) = − w(z) = Q 2π(z − z0 )

(IV.42a) (IV.42b) has a simple pole at z = z0 . Using polar coordinates (r, θ) centered on that pole, the flow velocity is purely radial: Q ~v(r, θ) = ~er (IV.42c) 2πr as displayed in the left panel of Fig. IV9 while the flow potential and the stream function are ϕ(r, θ) = − Q log r, 2π ψ(r, θ) = − Q θ. 2π (IV.42d) By computing the flux of velocity through a closed curve circling the polee.g a circle centered on z0 , which is an equipotential of ϕ, one finds that Q represents the mass flow rate through that curve. If Q is positive, there is a source of flow at z0 ; is Q is negative, there is a sink there, in which the fluid disappears. y y 6 6 - x - x Figure IV.9 – Streamlines (full) and equipotential lines (dashed) for a flow source (IV42c) (left) and a pointlike vortex (IV.43b) (right) (23) The reader unwilling to take the logarithm of a dimensionful quantityto which she is entirely entitledmay divide z − z0 resp. r by a length in the potentials

(IV42a) and (IV43a) resp (IV42d) and (IV43c), or write the difference in Eq. (IV45) as the logarithm of a quotient She will however quickly convince herself that this does not affect the velocities (IV.42b) and (IV43b), nor the potential (IV44a) 63 IV.4 Potential flows Pointlike vortex The “conjugate” flow to the previous one, i.e that for which ϕ and ψ are exchanged, corresponds to the complex potential(23) φ(z) = iΓ log(z − z0 ) with Γ ∈ R, z0 ∈ C. 2π (IV.43a) Using as above polar coordinates (r, θ) centered on z0 , the flow velocity is purely tangential, ~v(r, θ) = Γ ~eθ , 2πr2 (IV.43b) as shown in Fig. IV9 (right), where the basis vector ~eθ is normalized to r, cf Eq (C6) The complex potential (IV.43a) thus describes a vortex situated at z0 In turn, the velocity potential and stream function read ϕ(r, θ) = − Γ θ, 2π ψ(r, θ) = Γ log r, 2π (IV.43c) to be compared with those for a flow source, Eq. (IV42d) Remark: When writing down

the complex velocity potentials (IV.42a) or (IV43a), we left aside the issue of the (logarithmic!) branch point at z = z0 and we did not specify which branch of the logarithm we consider. Now, either potential corresponds to a flow that is actually defined on a doubly connected region, since the velocity diverges at z = z0 . From the discussion in § IV42 b, on such domains the potential is a multivalued object, yet this is irrelevant for the physical quantities, namely the velocity field, which remains uniquely defined at each point. This is precisely what is illustrated here by the different branches of the logarithm, which differ by a constant multiple of 2iπ that does not affect the derivative. Flow dipole A further possible irrotational and incompressible two-dimensional flow is that defined by the complex potential µ eiα φ(z) = with µ ∈ R, α ∈ R, z0 ∈ C (IV.44a) z − z0 leading to the complex flow velocity w(z) = µ eiα . (z − z0 )2 (IV.44b) Again, both φ(z)

and w(z) are singular at z0 . Using polar coordinates (r, θ) centered on z0 , the flow velocity reads µ µ ~v(r, θ) = 2 cos(θ − α)~er + 3 sin(θ − α)~eθ , (IV.44c) r r which shows that the angle α gives the overall orientation of the flow with respect to the x-direction. Setting for simplicity α = 0 and coming back momentarily to Cartesian coordinates, the flow potential and stream function corresponding to Eq. (IV44a) are µx µy ϕ(x, y) = 2 , ψ(x, y) = − 2 . (IV.44d) 2 x +y x + y2 Thus, the streamlines are the curves x2 + y 2 = const. × y, ie they are circles centered on the y-axis and tangent to the x-axis, as represented in Fig. IV10, where everything is tilted by an angle α One can check that the flow dipole (IV.44a) is actually the superposition of a pair of infinitely close source and sink with the same mass flow rate in absolute value:

 µ φ(z) = lim log z − z0 + ε e−iα − log z − z0 − ε e−iα . (IV.45) ε0 2ε 64 Non-relativistic

flows of perfect fluids y 6 α - x Figure IV.10 – Streamlines for a flow dipole (IV44a) centered on the origin This is clearly fully analogous to an electric dipole potential being the superposition of the potentials created by electric charges +q and −qand justifies the denomination “dipole flow”. One can similarly define higher-order multipoles: flow quadrupoles, octupoles, . , for which the order of the pole of the velocity at z0 increases (order 1 for a source or a sink, order 2 for a dipole, order 3 for a quadrupole, and so on). Remarks: ∗ The complex flow potentials considered until nownamely those of uniform flows (IV.41), sources or sinks (IV.42a), pointlike vortices (IV43a), and dipoles (IV44a) or multipolesand their superpositions are the only two-dimensional flows valid on an unbounded domain As a matter of fact, demanding that the flow velocity ~v(~r) be uniform at infinity and that the complex velocity w(z) be analytic except at a finite number of

singularitiessay only one, at z0 , to simplify the argumentation, then w(z) may be expressed as a superposition of integer powers of 1/(z − z0 ): ∞ X a−p w(z) = , (IV.46a) (z − z0 )p p=0 since any positive power of (z − z0 ) would be unbounded when |z| ∞. Integrating over z, see Eq. (IV40), the allowed complex potentials are of the form ∞ X p a−p−1 φ(z) = −a0 z − a−1 log(z − z0 ) + . (IV.46b) (z − z0 )p p=1 ∗ Conversely, the reader can checkby computing the integral of w(z) along a contour at infinity that the total mass flow rate and circulation of the velocity field for a given flow are respectively the real and imaginary parts of the residue a−1 in the Laurent series of its complex velocity w(z), i.e are entirely governed by the source/sink term (IV42a) and vortex term (IV43a) in the complex potential. ∗ Eventually, the singularities that arise in the flow velocity will in practice not be a problem, since these points will not be part of the

physical flow, as we shall see on an example in § IV.43 c Flow inside or around a corner As a last example, consider the complex flow potential φ(z) = A e−iα (z − z0 )n 1 with A ∈ R, α ∈ R, n ≥ , z0 ∈ C. 2 (IV.47a) 65 IV.4 Potential flows Figure IV.11 – Streamlines for the flow defined by potential (IV47a) with from top to bottom and from left to right n = 3, 32 , 1, 34 , 35 and 21 . Except in the case n = 1, this potential cannot represent a flow on an unbounded domain, since one easily checks that the velocity is unbounded as |z| goes to infinity. The interest of this potential lies rather the behavior in the vicinity of z = z0 . As a matter, writing down the flow potential and the stream function in a system of polar coordinates centered on z0 , ϕ(r, θ) = A rn cos(nθ − α), ψ(r, θ) = A rn sin(nθ − α) (IV.47b) shows that they both are (π/n)-periodic functions of the polar angle θ. Thus the flow on the domain D delimited by the streamlines

ψ(r, α) and ψ(r, α + π/n) is isolated from the motion in the remainder of the complex plane. One may therefore assume that there are walls along these two streamlines, and that the complex potential (IV.47a) describes a flow between them For n = 1, one recovers the uniform flow (IV.41)in which we are free to put a wall along any streamline, restricting the domain D to a half plane instead of the whole plane. If n > 1, π/n is smaller than π and the domain D is comprised between a half-plane; in that case, the fluid motion is a flow inside a corner. On the other hand, for 12 ≤ n < 1, π/n > π, so that the motion is a flow past a corner. The streamlines for the flows obtained with six different values for n are displayed in Fig. IV11, namely two flows in corners with angles π/3 and 2π/3, a uniform flow in the upper half plane, two flows past corners with inner angles 2π/3 and π/3, and a flow past a flat plaque, corresponding respectively to n = 3, 23 , 1, 43 , 35

and 21 . IV.43 c Two-dimensional flows past a cylinder ::::::::::::::::::::::::::::::::::::::::::::::: Thanks to the linearity of the Laplace differential equations, one may add “elementary” solutions of the previous paragraph to obtain new solutions, which describe possible two-dimensional flows. We now present two examples, which represent flows coming from infinity, where they are uniform, and falling on a cylindereither immobile or rotating around its axis. Acyclic flow Let us superpose the complex potentials for a uniform flow (IV.41) along the x-direction and a flow dipole (IV.44a) situated at the origin and making an angle α = π with the vector ~ex :   R2 φ(z) = −v∞ z + , (IV.48a) z where the dipole strength µ was written as R2 v∞ . Adopting polar coordinates (r, θ), this ansatz 66 Non-relativistic flows of perfect fluids y 6 - x Figure IV.12 – Streamlines for the acyclic potential flow past a cylinder (IV48a) leads to the velocity potential and

stream function     R2 R2 cos θ, ψ(r, θ) = −v∞ r − sin θ. ϕ(r, θ) = −v∞ r + r r (IV.48b) One sees that the circle r = R is a line of constant ψ, i.e a streamline This means that the flow outside that circle is decoupled from that inside. In particular, one may assume that the space inside the circle is filled by a solid obstacle, a “cylinder”,(24) without changing the flow characteristics on R2 deprived from the disk r < R. The presence of this obstacle has the further advantage that it “hides” the singularity of the potential or the resulting velocity at z = 0, by cleanly removing it from the domain over which the flow is defined. This is illustrated, together with the streamlines for this flow, in Fig. IV12 From the complex potential (IV.48a) follows at once the complex velocity   R2 w(z) = v∞ 1 − 2 , (IV.49a) z which in polar coordinates gives  ~v(r, θ) = v∞ R2 1− 2 r     R2 ~eθ cos θ~er − 1 + 2 sin θ . r r (IV.49b) The

latter is purely tangential for r = R, in agreement with the fact that the cylinder surface is a streamline. The flow velocity even fully vanishes at the points with r = R and θ = 0 or π, which are thus stagnation points.(liii) Assuming that the motion is stationary, one can calculate the force exerted on the cylinder by the flowing fluid. Invoking the Bernoulli equation (IV11)which holds since the flow is steady and incompressibleand using the absence of vorticity, which leads to the constant being the same throughout the flow, one obtains 1 2 1 2 2 , P (~r) + ρ~v(~r)2 = P ∞ + ρv∞ (24) The denomination is motivated by the fact that even though the flow characteristics depend on two spatial coordinates only, the actual flow might take in place in a three-dimensional space, in which case the obstacle is an infinite circular cylinder. (liii) Staupunkte 67 IV.4 Potential flows where P ∞ denotes the pressure at infinity. That is, at each point on the surface of the

cylinder 1  2 1 2 2 2 1 − 4 sin2 θ , P (R, θ) = P ∞ + ρ v∞ −~v(R, θ)2 = P ∞ + ρv∞ 
where the second identity follows from Eq. (IV49b) The resulting stress vector on the vector at a given θ is directed radially towards the cylinder center, T~s (R, θ) = −P (R, θ)~er (R, θ). Integrating over θ ∈ [0, 2π], the total force on the cylinder due to the flowing fluid simply vanishesin conflict with the intuition, phenomenon which is known as d’Alembert paradox .(w) The intuition according to which the moving fluid should exert a force on the immobile obstacle is good. What we find here is a failure of the perfect-fluid model, which is in that case too idealized, by allowing the fluid to slip without friction along the obstacle. Cyclic flow To the flow profile which was just considered, we add a pointlike vortex (IV.43a) situated at the origin   R2 iΓ z + φ(z) = −v∞ z + log , (IV.50a) z 2π R where we have divided z by R in the logarithm to have a

dimensionless argument, although this plays no role for the velocity. Comparing with the acyclic flow, which models fluid motion around a motionless cylinder, the complex potential may be seen as a model for the flow past a rotating cylinder, as in the case of the Magnus effect (§ IV.22 d) Adopting polar coordinates (r, θ), the velocity potential and stream function read     R2 Γ Γ r R2 cos θ − θ, ψ(r, θ) = −v∞ r − sin θ + log , (IV.50b) ϕ(r, θ) = −v∞ r + r 2π r 2π R so that the circle r = R remains a streamline, delimiting a fixed obstacle. The resulting velocity field reads in complex form   R2 iΓ w(z) = v∞ 1 − 2 − , z 2πz (IV.51a) and in polar coordinates  ~v(r, θ) = v∞ 1− R2 r2     R2 ~eθ Γ cos θ~er − 1 + 2 − sin θ . r 2πrv∞ r (IV.51b) The latter is purely tangential for r = R, in agreement with the fact that the cylinder surface is a streamline. One easily checks that when the strength of the vortex is not too

large, namely Γ ≤ 4πR v∞ , the flow has stagnations points on the surface of the cylindertwo if the inequality holds in the strict sense, a single degenerate point if Γ = 4πR v∞ , as illustrated in Fig. IV13 If Γ > 4πR v∞ , the flow defined by the complex potential (IV.50a) still has a stagnation point, yet now away from the surface of the rotating cylinder, as exemplified in Fig. IV14 In either case, repeating the same calculation based on the Bernoulli equation as for the acyclic flow allows one to derive the force exerted by the fluid on the cylinder. The resulting force no longer vanishes, but equals −Γρv∞ ~ey on a unit length of the cylinder, where ρ is the mass density of the fluid and ~ey the unit basis vector in the y-direction. This is in line with the arguments presented in § IV.22 d (w) J. le Rond d’Alembert, 1717–1783 68 Non-relativistic flows of perfect fluids Figure IV.13 – Streamlines for the cyclic potential flow past a (rotating)

cylinder (IV50a) with Γ/(4πR v∞ ) = 0.25 (left) or 1 (right) Figure IV.14 – Streamlines for the cyclic potential flow past a (rotating) cylinder (IV50a) with Γ/(4πR v∞ ) = 4. IV.43 d Conformal deformations of flows ::::::::::::::::::::::::::::::::::::::::: A further possibility to build two-dimensional potential flows is to “distort” the elementary solutions of § IV.43 b, or linear combinations of these building blocks Such deformations may however not be arbitrary, since they must preserve the orthogonality at each point in the fluid of the streamline (with constant ψ) and the equipotential line (constant ϕ) passing through that point. Besides rotations and dilationswhich do not distort the profile of the solution, and are actually already taken into account in the solutions of § IV.43 b, the generic class of transformations of the (complex) plane that preserve angles locally is that of conformal maps. As recalled in Appendix D.4, such conformal mappingsbetween

open subsets of the complex planes of variables z and Zare defined by any holomorphic function Z = f (z) whose derivative is everywhere non-zero and by its inverse F . If φ(z) denotes an arbitrary complex flow potential on the z-plane, then Φ(Z) ≡ φ(F (Z)) is a flow potential on the Z-plane. Applying the chain rule, the associated complex flow velocity is w(F (Z))F 0 (Z), where F 0 denotes the derivative of F . A first example is to consider the trivial uniform flow with potential φ(z) = A z, and the conformal mapping z 7 Z = f (z) = z 1/n with n ≥ 12 . The resulting complex flow potential on the Z-plane is Φ(Z) = −A Z n . 69 IV.4 Potential flows Except in the trivial case n = 1, f (z) is singular at z = 0, where f 0 vanishes, so that the mapping is non-conformal: cutting a half-line ending at z = 0, f maps the complex plane deprived from this half-line onto an angular sector delimited by half-lines making an angle π/nas already seen in § IV.43 b Joukowsky transform

A more interesting set of conformally deformed fluid flows consists of those provided by the use of the Joukowsky transform (x) R2 Z = f (z) = z + J (IV.52) z where RJ ∈ R. The mapping (IV.52) is obviously holomorphic in the whole complex z-plane deprived of the originwhich a single pole, and has 2 points z = ±RJ at which f 0 vanishes. These two singular points correspond in the Z-plane to algebraic branch points of the reciprocal function z = F (Z) at Z = ±2RJ . To remove them, one introduces a branch cut along the line segment |X| ≤ 2RJ On the open domain U consisting complex Z-plane deprived from that line segment, F is holomorphic and conformal. One checks that the cut line segment is precisely the image by f of the circle |z| = RJ in the complex z-plane. Thus, f and F provide a bijective mapping between the exterior of the circle |z| = RJ in the z-plane and the domain U in the Z-plane. Another property of the Joukowsky transform is that the singular points z = ±RJ are

zeros of f 0 of order 1, so that angles are locally multiplied by 2. That is, every continuously differentiable curve going through z = ±RJ is mapped by f on a curve through Z = ±2RJ with an angular point, i.e a discontinuous derivative, there Consider first the circle C (0, R) in the z-plane of radius R > RJ centered on the origin; it can be parameterized as  C (0, R) = z = R eiϑ , 0 ≤ ϑ ≤ 2π . Its image in the Z-plane by the Joukowsky transform (IV.52) is the set of points such that     RJ2 RJ2 cos ϑ + i R − sin ϑ, 0 ≤ ϑ ≤ 2π, Z = R+ R R that is, the ellipse centered on the origin Z = 0 with semi-major resp. semi-minor axis R + RJ2 /R resp. R−RJ2 /R along the X- resp Y -direction Accordingly, the flows past a circular cylinder studied in § IV.43 c can be deformed by f into flows past elliptical cylinders, where the angle between the ellipse major axis and the flow velocity far from the cylinder may be chosen at will. Bibliography for Chapter IV •

National Committee for Fluid Mechanics film & film notes on Vorticity; • Faber [1] Chapters 1.7, 28–29, 41–412; • Feynman [8, 9] Chapter 40; • Guyon et al. [2] Chapters 53–54, 61–63, 65–66 & 7-1–73; • Landau–Lifshitz [3, 4] Chapter I § 3, 5, 8–11; • Sommerfeld [5, 6] Chapters II § 6,7 and IV § 18,19. (x) N. E ukovski = N. E Zhukovsky, 1847–1921 C HAPTER V Waves in non-relativistic perfect fluids A large class of solutions of the equations of motion (III.9), (III18) and (III33) is that of waves Quite generically, this denomination designates “perturbations” of some “unperturbed” fluid motion, which will also be referred to as background flow. In more mathematical terms, the starting point is a set of fields {ρ0 (t,~r),~v0 (t,~r), P 0 (t,~r)} solving the equations of motion, representing the background flow. The wave then consists of a second set of fields {δρ(t,~r), δ~v(t,~r), δ P (t,~r)} which are added the background ones,

such that the resulting fields ρ(t,~r) = ρ0 (t,~r) + δρ(t,~r), (V.1a) P (t,~r) = P 0 (t,~r) + δ P (t,~r), (V.1b) ~v(t,~r) =~v0 (t,~r) + δ~v(t,~r) (V.1c) are solutions to the equations of motion. Different kinds of perturbationstriggered by some source which will not be specified hereafter, and is thus to be seen as an initial conditioncan be considered, leading to different phenomena. A first distinction, with which the reader is probably already familiar, is that between traveling waves, which propagate, and standing waves, which do not. Mathematically, in the former case the propagating quantity does not depend on space and time independently, but rather on a combination like (in a one-dimensional case) x − cϕ t, some propagation speed. In contrast, in standing waves the space and time dependence of the “propagating” quantity factorize. Hereafter, we shall mostly mention traveling waves. Another difference is that between “small” and “large” perturbations

or, in more technical terms, between linear and nonlinear waves. In the former case, which is that of sound waves (Sec V1) or the simplest gravity-controlled surface waves in liquids (Sec. V31), the partial differential equation governing the propagation of the wave is linearwhich means that nonlinear terms have been neglected. Quite obviously, nonlinearities of the dynamical equationsas eg the Euler equation are the main feature of nonlinear waves, as for instance shock waves (V.2) or solitons (Sec V32) V.1 Sound waves By definition, the phenomenon which in everyday life is referred to as “sound” consists of small adiabatic pressure perturbations around a background flow, where adiabatic actually means that the entropy remains constant. In the presence of such a wave, each point in the fluid undergoes alternative compression and rarefaction processes. That is, these waves are by construction (parts of) a compressible flow. We shall first consider sound waves on a uniform perfect

fluid at rest (Sec. V11) What then? Doppler effect? Riemann problem? 71 V.1 Sound waves V.11 Sound waves in a uniform fluid at rest Neglecting the influence of gravity, a trivial solution of the dynamical equations of perfect fluids is that with uniform and time independent mass density ρ0 and pressure P 0 , with a vanishing flow velocity ~v0 = ~0. Assuming in addition that the particle number N0 in the fluid is conserved, its entropy has a fixed value S0 . These conditions will represent the background flow we consider here Inserting the values of the various fields in relations (V.1), a perturbation of this background flow reads ρ(t,~r) = ρ0 + δρ(t,~r), (V.2a) P (t,~r) = P 0 + δ P (t,~r), (V.2b) ~v(t,~r) = ~0 + δ~v(t,~r). (V.2c) The necessary “smallness” of perturbations means for the mass density and pressure terms |δρ(t,~r)|  ρ0 , |δ P (t,~r)|  P 0 . (V.2d) Regarding the velocity, the background flow does not explicitly specify a reference scale, with

which the perturbation should be compared. As we shall see below, the reference scale is actually implicitly contained in the equation(s) of state of the fluid under consideration, and the condition of small perturbation reads |δ~v(t,~r)|  cs (V.2e) with cs the speed of sound in the fluid. Inserting the fields (V.2) in the equations of motion (III9) and (III18) and taking into account the uniformity and stationarity of the background flow, one finds   ∂δρ(t,~r) ~ · δ~v(t,~r) + ∇ ~ · δρ(t,~r) δ~v(t,~r) = 0, (V.3a) + ρ0 ∇ ∂t     ∂δ~v(t,~r)   ~ ~ P (t,~r) = 0. ρ0 + δρ(t,~r) + δ~v(t,~r) · ∇ δ~v(t,~r) + ∇δ (V.3b) ∂t The required smallness of the perturbations will help us simplify these equations, in that we shall only keep the leading-order terms in an expansion in which we consider ρ0 , P 0 as zeroth-order quantities while δρ(t,~r), δ P (t,~r) and δ~v(t,~r) are small quantities of first order. Accordingly, the third term in the continuity

equation is presumably much smaller than the other two, and may be left aside in a first approximation. Similarly, the contribution of δρ(t,~r) and the convective term within the curly brackets on the left hand side of Eq. (V3b) may be dropped The equations describing the coupled evolutions of δρ(t,~r), δ P (t,~r) and δ~v(t,~r) are thus linearized : ∂δρ(t,~r) ~ · δ~v(t,~r) = 0, (V.4a) + ρ0 ∇ ∂t ∂δ~v(t,~r) ~ ρ0 + ∇δ P (t,~r) = 0. (V.4b) ∂t To have a closed system of equations, we still need a further relation between the perturbations. This will be provided by thermodynamics, i.e by the implicit assumption that the fluid at rest is everywhere in a state in which its pressure P is function of mass density ρ, (local) entropy S, and (local) particle number N , i.e that there exists a unique relation P = P (ρ, S, N ) which is valid at each point in the fluid and at every time. Expanding this relation around the (thermodynamic) point corresponding to the

background flow, namely P 0 = P (ρ0 , S0 , N0 ), one may write      
∂P ∂P ∂P P ρ0 + δρ, S0 + δS, N0 + δN = P 0 + δρ + δS + δN, ∂ρ S,N ∂S ρ,N ∂N S,ρ where the derivatives are taken at the point (ρ0 , S0 , N0 ). Here, we wish to consider isentropic 72 Waves in non-relativistic perfect fluids perturbations at constant particle number, i.e δS and δN vanish, leaving   ∂P δP = δρ. ∂ρ S,N For this derivative, we introduce the notation c2s   ∂P ≡ ∂ρ S,N (V.5) where both sides actually depend on ρ0 , S0 and N0 . One may then express δ P as function of δρ, ~ P (t,~r) by c2s ∇δρ(t,~ ~ and replace ∇δ r) in Eq. (V4b) The resulting equations for δρ(t,~r) and δ~v(t,~r) are linear first order partial differential equations. Thanks to the linearity, their solutions form a vector spaceat least as long as no initial condition has been specified. One may for instance express the solutions as Fourier transforms, ie

superpositions of plane waves. Accordingly, we test the ansatz ~ ev(ω, ~k) e−iωt+i~k·~r , δ~v(t,~r) = δ~ e ~k) e−iωt+ik·~r , δρ(t,~r) = δρ(ω, (V.6) ev that a priori depend on ω and ~k and are determined by the initial e δ~ with respective amplitudes δρ, conditions for the problem. In turn, ω and ~k are not necessarily independent from each other With this ansatz, Eqs. (V4) become ev(ω, ~k) = 0 e ~k) + iρ0 ~k · δ~ −iω δρ(ω, ev(ω, ~k) + ic2 ~k δρ(ω, e ~k) = 0. −iωρ0 δ~ s (V.7a) (V.7b) ev(ω, ~k) is proportional to ~k; in particular, it lies along From the second equation, the amplitude δ~ ev simply equals the product of the norms of the the same direction. That is, the inner product ~k · δ~ two vectors. Omitting from now on the (ω, ~k)-dependence of the amplitudes, the inner product of Eq. (V7b) with ~kwhich does not lead to any loss of informationallows one to recast the system as ! ! ! e 0 −ω ρ0 δρ . = ev ~k · δ~ c2s~k 2

−ωρ0 0 ev = ~0, i.e the absence of any perturbation In e = 0, δ~ A first, trivial solution to this system is δρ order for non-trivial solutions to exist, the determinant (ω 2 − c2s~k 2 )ρ0 of the system should vanish. This leads at once to the dispersion relation ω = ±cs |~k|. (V.8) Denoting by ~e~k the unit vector in the direction of ~k, the perturbations δρ(t,~r) and δ~v(t,~r) defined by Eq. (V6), as well as δ P (t,~r) = c2s δρ(t,~r), are all functions of cs t ±~r ·~e~k These are thus traveling waves,(liv) , that propagate with the phase velocity ω(~k)/|~k| = cs , which is independent of ~k. That is, cs is the speed of sound . For instance, for air at T = 300 K, the speed of sound is cs = 347 m · s−1 . Air is a diatomic ideal gas, i.e it has pressure P = N kB T /V and internal energy U = 25 N kB T         ∂P V 2 ∂P V2 N kB T N kB ∂T This then gives c2s = =− =− − + . ∂ρ S,N mN ∂V S,N mN V2 V ∂V S,N (liv) fortschreitende Wellen

73 V.1 Sound waves The thermodynamic relation dU = T dS − P dV + µ dN yields at constant entropy and particle number       5 ∂T 2P ∂U ∂T 2 N kB T = − N kB i.e N kB =− P =− =− . ∂V S,N 2 ∂V S,N ∂V S,N 5 5 V leading to c2s = 7 kB T , with mair = 29/NA g · mol−1 . 5 mair  Remarks: ∗ Taking the real parts of the complex quantities in the harmonic waves (V.6), so as to obtain real-valued δρ, δ P and δ~v, one sees that these will be alternatively positive and negative, and in averageover a duration much longer than a period 2π/ωzero. This in particular means that the successive compression and condensation (δ P > 0, δρ > 0) or depression and rarefaction(lv) (δ P < 0, δρ < 0) processes do not lead to a resulting transport of matter. ∗ A single harmonic wave (V.6) is a traveling wave Yet if the governing equation or systems of equations is linear or has been linearized, as was done here, the superposition of harmonic waves is

a valid solution. In particular, the superposition of two harmonic traveling waves with equal frequencies ω, opposite waves vectors ~kwhich is allowed by the dispersion relation (V.8) and equal amplitudes leads to a standing wave, in which the dependence on time and space is proportional to eiωt cos(~k · ~r). ev(ω, ~k) and ~k means that the sound waves Coming back to Eq. (V7b), the proportionality of δ~ in a fluid are longitudinal in contrast to electromagnetic waves in vacuum, which are transversal waves. The nonexistence of transversal waves in fluids reflects the absence of forces that would act against shear deformations so as to restore some equilibrium shapeshear viscous effects cannot play that role. In contrast, there can be transversal sound waves in elastic solids, as e.g the so-called S-modes (shear modes) in geophysics. The inner product of Eq. (V7b) with ~k, together with the dispersion relation (V8) and the ev and ~k, leads to the relation collinearity of δ~ ev e

δ~ δρ ev = c2 ~k δρ e = ⇔ ωρ0 ~k δ~ s cs ρ0 for the amplitudes of the perturbations. This justifies condition (V2e), which is then consistent with (V.2d) Similarly, inserting the ansatz (V6) in Eq (V3b), the terms within curly brackets
ev + i ~k · δ~ ev δ~ ev: again, neglecting the second with respect to the first is equivalent to become −iω δ~ e requesting δ~v  cs . Remark: Going back to Eqs. (V4), the difference of the time derivative of the first one and the ~ P has been replaced by c2 ∇ρleads ~ divergence of the second onein which ∇ to the known wave s (25) equation ∂ 2ρ(t,~r) (V.9a) − c2s 4ρ(t,~r) = 0, ∂t2 If the flowincluding the background flow on which the sound wave develops, in case ~v0 is not ~ trivial as it is hereis irrotational, so that one may write ~v(t,~r) = −∇ϕ(t,~ r), then the velocity potential ϕ also obeys the same equation ∂ 2 ϕ(t,~r) − c2s 4ϕ(t,~r) = 0. ∂t2 (25) (lv) This traditional denomination is totally out

of place in a chapter in which there are several types of waves, each of which has its own governing “wave equation”. Yet historically, due to its role for electromagnetic or sound waves, it is the archetypal wave equation, while the equations governing other types of waves often have a specific name. Verdünnung 74 V.12 Sound waves on moving fluids V.13 Riemann problem Rarefaction waves Waves in non-relativistic perfect fluids 75 V.2 Shock waves V.2 Shock waves When the amplitude of the perturbations considered in Sec. (V1) cannot be viewed as small, as for instance if |δ~v|  cs does not hold, then the linearization of the equations of motion (V.3) is no longer licit, and the nonlinear terms play a role. A possibility is then that at a finite time t in the evolution of the fluid, a discontinuity in some of the fields may appear, referred to as shock wave.(lvi) How this may arise will be discussed in the case of a one-dimensional problem (Sec. (V21)) At a

discontinuity, the differential formulation of the conservation laws derived in Chap. III no longer holds, and it becomes necessary to study the conservation of mass, momentum and energy across the surface of discontinuity associated with the shock wave (Sec. V22) V.21 Formation of a shock wave in a one-dimensional flow As in Sec. (V11), we consider the propagation of an adiabatic perturbation of a background fluid at rest, neglecting the influence of gravity or other external volume forces. In the one-dimensional case, the dynamical equations (V.3) read ∂ρ(t, x) ∂δv(t, x) ∂ρ(t, x) + ρ(t, x) + δv(t, x) = 0, ∂t ∂x ∂x   ∂δv(t, x) ∂δ P (t, x) ∂δv(t, x) + δv(t, x) = 0. + ρ(t, x) ∂t ∂x ∂x (V.10a) (V.10b) The variation of the pressure δ P (t, x) can again be expressed in terms of the variation in the mass density δρ(t, x) by invoking a Taylor expansion [cf. the paragraph between Eqs (V4) and (V5)] Since the perturbation of the background “flow”

is no longer small, the thermodynamic state around which this Taylor expansion is performed is not necessarily that corresponding to the unperturbed fluid, but rather an arbitrary state, so that δ P (t, x) cs (ρ)2 δρ(t, x), where the speed of sound is that in the perturbed flow. When differentiating this identity, the derivative of δρ(t, x) with respect to x is also the derivative of ρ(t, x), since the unperturbed fluid state is uniform. Accordingly, one may recast Eqs (V10) as ∂δv(t, x) ∂ρ(t, x) ∂ρ(t, x) + ρ(t, x) + δv(t, x) = 0, ∂t ∂x ∂x   ∂δv(t, x) ∂ρ(t, x) ∂δv(t, x) ρ(t, x) + δv(t, x) + cs (ρ)2 = 0, ∂t ∂x ∂x (V.11a) (V.11b) which constitutes a system of two coupled partial differential equations for the two unknown fields ρ(t, x) and δv(t, x) = v(t, x). To tackle these equations, one may assume that the mass density and the flow velocity have parallel dependences on time and spaceas suggested by the fact that this property holds in

the linearized case of sound waves, in which both ρ(t,~r) and ~v(t,~r) propagate with the same phase (cs |~k|t +
~k · ~r). Thus, the dependence of v on t and x is replaced with a functional dependence v ρ(t, x) , with the known value v(ρ0 ) = 0 corresponding to the unperturbed fluid at rest. Accordingly, the partial derivatives of the flow velocity with respect to t resp x become ∂v(t, x) dv(ρ) ∂ρ(t, x) = ∂t dρ ∂t resp. ∂v(t, x) dv(ρ) ∂ρ(t, x) = . ∂x dρ ∂x The latter identities may then be inserted in Eqs. (V11) If one further multiplies Eq (V11a) by (lvi) Stoßwelle 76 Waves in non-relativistic perfect fluids ρ(t, x) dv(ρ)/dρ and then subtracts Eq. (V11b) from the result, there comes    2 2 ∂ρ(t, x) 2 dv(ρ) − cs (ρ) ρ = 0, dρ ∂x that is, discarding the trivial solution of a uniform mass density, dv(ρ) cs (ρ) =± . dρ ρ (V.12) Under the simultaneous replacements v −v, x −x, cs −cs , equations (V.11)-(V12) remain

invariant. Accordingly, one may restrict the discussion of Eq (V12) to the case with a + signthe − case amounts to considering a wave propagating in the opposite direction with the opposite velocity. The flow velocity is then formally given by Z ρ cs (ρ0 ) 0 v(ρ) = dρ , 0 ρ0 ρ where we used v(ρ0 ) = 0, while Eq. (V11b) can be rewritten as

 ∂ρ(t, x) ∂ρ(t, x)  + v ρ(t, x) + cs ρ(t, x) = 0. ∂t ∂x (V.13) Assuming that the mass density perturbation propagates as a traveling wave, i.e making the ansatz δρ(t, x) ∝ f (x − cw t) in Eq. (V13), then its phase velocity cw will be given by cw = cs (ρ) + v Invoking Eq. (V12) then shows that dv(ρ)/dρ > 0, so that cw grows with increasing mass density: the denser regions in the fluid will propagate faster than the rarefied ones and possibly catch up with themin case the latter where “in front” of the propagating perturbationas illustrated in Fig. V1 In particular, there may arise after a finite amount of

time a discontinuity of the function ρ(t, x) at a given point x0 . The (propagating) point where this discontinuity takes place represents the front of a shock wave. ρ6 t0 - x t1 > t0 - x t2 > t1 - x t3 > t2 - x t4 > t3 - x Figure V.1 – Schematic representation of the evolution in time of the spatial distribution of dense and rarefied regions leading to a shock wave. 77 V.2 Shock waves V.22 Jump equations at a surface of discontinuity To characterize the properties of a flow in the region of a shock wave, one needs first to specify the behavior of the physical quantities of relevance at the discontinuity, which is the object of this Section. Generalizing the finding of the previous Section in a one-dimensional setup, in which the discontinuity arises at a single (traveling) point, in the three-dimensional case there will be a whole surface of discontinuity,(lvii) that propagates in the unperturbed background fluid. For the sake of brevity, the

dependence on t and ~r of the various fields of interest will be omitted. To describe the physics at the front of the shock wave, we adopt a comoving reference frame R, which moves with the surface of discontinuity, and in this reference frame we consider a system of Cartesian coordinates (x1 , x2 , x3 ) with the basis vector ~e1 perpendicular to the propagating surface. The region in front resp. behind the surface will be denoted by (+) resp (−); that is, the fluid in which the shock waves propagates flows from the (+)- into the (−)-region. The jump(lviii) of a local physical quantity g (~r) across the surface of discontinuity is defined as   g ≡ g+ − g− , (V.14) where g+ resp. g− denotes the limiting value of g as x1 0+ resp x1 0− In case such a local quantity is actually continuous at the surface of discontinuity, then its jump across the surface vanishes. At a surface of discontinuity Sd , the flux densities of mass, momentum, and energy across the surface, i.e

along the x1 -direction, must be continuous, so that mass, momentum, and energy remain locally conserved. These requirements are expressed by the jump equations (lix)  1  ρ v = 0, (V.15a)  i1  T = 0 ∀i = 1, 2, 3, (V.15b)    1 2 ρ~v + e + P v1 = 0, (V.15c) 2 where the momentum flux density tensor has components T ij = P g ij + ρ vi vj [see Eq. (III21b)], with g ij = δ ij in the case of Cartesian coordinates. The continuity of the mass flux density across the surface of discontinuity (V.15a) can be recast as (ρv1 )− = (ρv1 )+ ≡ j1 . (V.16) A first, trivial solution arises if there is no flow of matter across surface Sd , i.e if (v1 )+ = (v1 )−  = 0. In that case, Eq. (V15c) is automatically satisfied Condition (V15b) for i = 1 becomes P = 0, i.e the pressure is the same on both sides of Sd Eventually, Eq (V15b) with i = 2 or 3 holds automatically. All in all, there is no condition on the behavior of ρ, v2 or v3 across the surface of discontinuitywhich

means that these quantities may be continuous or not, in the latter case with an arbitrary jump. If j1 does not vanish, that is if matter does flow Sd , then  across    the jump equation for the component T 21 = ρv2 v1 resp. T 31 = ρv3 v1 leads to v2 = 0 resp v3 = 0, ie the component v2 resp. v3 is continuous across the surface of discontinuity: (v2 )− = (v2 )+ resp. (v3 )− = (v3 )+ In turn, rewriting the jump equation for T 11 = P + ρ(v1 )2 with the help of j1 yields    1  1 1 1 2 P − − P + = j1 (v )+ − (v )− = j1 − . ρ+ ρ− (lvii) Unstetigkeitsfläche (lviii) Sprung (lix) Sprunggleichungen (V.17) (V.18) 78 Waves in non-relativistic perfect fluids Thus if ρ+ < ρ− , i.e if the fluid is denser in the (−)-region “behind” the shock frontas is suggested by Fig. V1, yet still needs to be proved, then P − > P + , while relation (V16) yields (v1 )+ > (v1 )− Conversely, ρ+ > ρ− leads to P − < P + and (v1 )+

< (v1 )− . One can show that the former case actually holds. Combining Eqs. (V16) and (V18) yields  (v1 )+ 2 = P − − P + ρ− ρ + P − − P + ρ− j12 = = ρ− − ρ+ ρ2+ ρ− − ρ+ ρ+ ρ2+ and similarly  1  2 P − − P + ρ+ . (v )− = ρ− − ρ+ ρ− If the jumps in pressure and mass density are small, one can show that their ratio is approximately the derivative ∂ P /∂ρ, here at constant entropy and particle number, i.e    1  2 ρ+ 2  1 2 ρ− 2 ρ− ∂P (v )− = cs , c . (v )+ ∂ρ S,N ρ+ ρ+ ρ− s With ρ− > ρ+ comes (v1 )+ > cs resp. (v1 )− < cs in front of resp behind the shock wave(26) The former identity means that an observer comoving with the surface of discontinuity sees in front a fluid flowing with a supersonic velocity, that is, going temporarily back to a reference frame bound to the unperturbed fluid, the shock wave moves with a supersonic velocity. Invoking the continuity across Sd of the product

ρ v1 and of the components v2 , v3 parallel to the surface of discontinuity, the jump equation (V.15c) for the energy flux density simplifies to     1 1 2 e+P j12 1 1 e+ + P + e− + P − − = 0. (v ) + = − 2 + 2 2 ρ 2 ρ+ ρ − ρ+ ρ− Expressing j12 with the help of Eq. (V18), one finds   P− − P+ 1 1 w− w+ + = − 2 ρ+ ρ− ρ− ρ+ with w = e + P the enthalpy density, or equivalently   e+ P− + P+ 1 1 e− = − − . 2 ρ+ ρ− ρ+ ρ− (V.19a) (V.19b) Either of these equations represents a relation between the thermodynamic quantities on both sides of the surface of discontinuity, and define in the space of the thermodynamic states of the fluid a so-called shock adiabatic curve, also referred to as dynamical adiabatic curve (lx) or Hugoniot(y) adiabatic curve, or Rankine(z) –Hugoniot relation. More generally, Eqs. (V16)–(V19) relate the dynamical fields on both sides of the surface of discontinuity associated with a shock wave, and constitute

the practical realization of the continuity conditions encoded in the jump equations (V.15) (26) (lx) (y) Here we are being a little sloppy: one should consider the right (x1 0+ ) and left (x1 0− ) derivatives, corresponding respectively to the (+) and (−)-regions, and thus find the associated speeds of sound (cs )+ and (cs )− instead of a single cs . dynamische Adiabate P. H Hugoniot, 1851–1887 (z) W. J M Rankine, 1820–1872 79 V.3 Gravity waves V.3 Gravity waves In this Section, we investigate waves that are “driven” by gravity, in the sense that the latter is the main force that acts to bring back the perturbed fluid to its unperturbed, “background” state. Such perturbations are generically referred to as gravity waves.(lxi) A first example is that of small perturbations at the free surface of a liquid originally at restthe “waves” of everyday language. In that case, some external source, as eg wind or an earthquake, leads to a local rise of the

fluid above its equilibrium level: gravity then acts against this rise and tends to bring back the liquid to its equilibrium position. In case the elevation caused by the perturbation is small compared to the sea depth, as well as in comparison to the perturbation wavelength, one has linear sea surface waves (Sec. V31) Another interesting case arises in shallow water, for perturbations whose horizontal extent is much larger than their vertical size, in which case one may find so-called solitary waves (Sec. V32) Throughout this Section, the flowscomprised of a background fluid at rest and the traveling perturbationare supposed to be two-dimensional, with the x-direction along the propagation direction and the z-direction along the vertical, oriented upwards so that the acceleration due to gravity is ~g = −g~ez . The origin z = 0 is taken at the bottom of the sea / ocean, which for the sake of simplicity is assumed to be flat. V.31 Linear sea surface waves A surface wave is a

perturbation of the altitudewith respect to the sea bottomof the free surface of the sea, which is displaced by an amount δh(t, x) from its equilibrium position h0 , where the latter corresponds to a fluid at rest, with a horizontal free surface. These variations in the position of the free surface signal the motion of the sea water, i.e a flow, with a corresponding flow velocity throughout the sea ~v(t, x, z). We shall model this motion as vorticity-free, which allows us to introduce a velocity potential ϕ(t, x, z), and assume that the mass density ρ of the sea water remains constant and uniform, i.e we neglect its compressibility. The sea is supposed to occupy an unbounded region of space, which is a valid assumption if one is far from any coast. V.31 a Equations of motion and boundary conditions :::::::::::::::::::::::::::::::::::::::::::::::::::::: Under the assumptions listed above, the equations of motion read [cf. Eq (IV30) and (IV31)] 2  ~ ∇ϕ(t, x, z) ∂ϕ(t, x, z) P

(t, x, z) − + + + gz = constant, (V.20a) ∂t 2 ρ where gz is the potential energy per unit mass of water, and   ∂2 ∂2 ϕ(t, x, z) = 0. + ∂x2 ∂z 2 (V.20b) To fully specify the problem, boundary conditions are still needed. As in the generic case for potential flow (Sec. IV4), these will be Neumann boundary conditions, involving the derivative of the velocity potential. • At the bottom of the sea, the water can have no vertical motion, corresponding to the usual impermeability condition; that is vz (z = 0) = − (lxi) Schwerewellen ∂ϕ ∂z = 0. z=0 (V.21a) 80 Waves in non-relativistic perfect fluids • At the free surface of the sea, the vertical component vz of the flow velocity equals the velocity of the surface, i.e it equals the rate of change of the position of the (material!) surface: Dδh(t, x) ∂ϕ(t, x, z) = − . ∂z Dt z=h0 +δh(t,x) Using D ∂ ∂ ∂ϕ ∂ ∂ = + vx = − , this gives Dt ∂t ∂x ∂t ∂x ∂x   ∂ϕ(t, x, z)

∂δh(t, x) ∂δh(t, x) ∂ϕ(t, x, z) + − = 0. ∂z ∂t ∂x ∂x z=h0 +δh(t,x) (V.21b) • At the free surface of the sea, the pressure on the water sideright below the surfaceis directly related to that just above the surface. The latter is assumed to be constant and equal at some value P 0 , which represents for instance the atmospheric pressure “at sea level”. As a first approximationwhose physical content will be discussed in the remark at the end of this paragraph, the pressure is equals on both sides of the sea surface:
P t, x, z = h0 +δh(t, x) = P 0 . (V.21c) Expressing the pressure with the help of Eq. (V20a), this condition may be recast as "  2 # ~ ∇ϕ(t, x, z) ∂ϕ(t, x, z) P0 − + + g δh(t, x) = − − gh0 + constant, (V.21d) ∂t 2 ρ z=h0 +δh(t,x) where the whole right hand side of the equation represents a new constant. Hereafter we look for solutions consisting of a velocity potential ϕ(t, x, z) and a surface profile δh(t, x), as

determined by Eqs. (V20) with conditions (V21) Remark: The assumption of an identical pressure on both sides of an interfaceeither between two immiscible liquids, or between a liquid and a gas, as hereis generally not warranted, unless the interface happens to be flat. If there is the least curvature, the surface tension associated with the interface will lead to a larger pressure inside the concavity of the interface. Neglecting this effect which we shall consider again in Sec. V32is valid only if the typical radius of curvature of the interface, which as we shall see below is the wavelength of the surface waves, is “large”, especially with respect to the deformation scale δh. V.31 b Harmonic wave assumption :::::::::::::::::::::::::::::::::::: Since the domain on which the wave propagates is unbounded, a natural ansatz for the solution of the Laplace equation (V.20b) is that of a harmonic wave ϕ(t, x, z) = f (z) cos(kx − ωt) (V.22) propagating in the x-direction with a

depth-dependent amplitude f (z). Inserting this form in the Laplace equation yields the linear ordinary differential equation d2f (z) − k 2 f (z) = 0, dz 2 whose obvious solution is f (z) = a1 ekz + a2 e−kz , with a1 and a2 two real constants. The boundary condition (V.21a) at the sea bottom z = 0 gives a1 = a2 , ie ϕ(t, x, z) = C cosh(kz) cos(kx − ωt), (V.23) with C a real constant. To make further progress with the equations of the system, and in particular to determine the profile of the free surface, further assumptions are needed, so as to obtain simpler equations. We shall now present a first such simplification, leading to linear waves. In Sec V32, another simplification of a more complicated started pointwill be considered, which gives rise to (analytically tractable!) nonlinear waves. 81 V.3 Gravity waves V.31 c Linear waves :::::::::::::::::::: As in the case of sound waves, we now assume that the perturbations are “small”, so as to be able to linearize

the equations of motion and those expressing boundary conditions. Thus, we shall ~ 2 is much smaller than |∂ϕ/∂t|, and that the displacement δh assume that the quadratic term (∇ϕ) of the free surface from its rest position is much smaller than the equilibrium sea depth h0 . To fix ideas the “swell waves” observed far from any coast on the Earth oceans or seas have a typical wavelength λ of about 100 m and an amplitude δh0 of 10 m or lessthe shorter the wavelength, the smaller the amplitude, while the typical sea/ocean depth h0 is 1–5 km. ~ 2  |∂ϕ/∂t| can on the one hand be made in Eq. (V20a), leading to The assumption (∇ϕ) − ∂ϕ(t, x, z) P (t, x, z) P0 + + gz = + gh0 , ∂t ρ ρ (V.24) in which the right member represents the zeroth order, while the left member also contains first order terms, which must cancel each other for the identity to hold. On the other hand, taking also into account the assumption |δh(t, x)|  h0 , the boundary conditions

(V.21b) and (V21d) at the free surface of the sea can be rewritten as ∂ϕ(t, x, z) ∂z and − ∂ϕ(t, x, z) ∂t ∂δh(t, x) =0 ∂t (V.25a) + g δh(t, x) = constant, (V.25b) + z=h0 z=h0 respectively. Together with the Laplace differential equation (V20b) and the boundary condition at the sea bottom (V.21a), the two equations (V25) constitute the basis of the Airy(aa) linear wave theory. Combining the latter two equations yields at once the condition  2  ∂ϕ(t, x, z) ∂ ϕ(t, x, z) = 0. +g ∂t2 ∂z z=h0 Using the velocity potential (V.23), this relation reads −ω 2 C cosh(kh0 ) cos(kx − ωt) + gkC sinh(kh0 ) cos(kx − ωt) = 0, resulting in the dispersion relation ω 2 = gk tanh(kh0 ). (V.26) This relation becomes even simpler in two limiting cases: • When kh0  1, or equivalently h0  λ where λ = 2π/k denotes the wavelength, which represents the case of gravity waves at the surface of deep sea,(27) then tanh(kh0 ) 1. In that case, the dispersion

relation simplifies to ω 2 = gk: the phase and group velocity of the traveling waves are r r ω dω(k) 1 g g cϕ = = , and cg = = k k dk 2 k respectively, both independent from the sea depth h0 . (27) The sea may not be “too deep”, otherwise the assumed uniformity of the water mass density along the vertical direction in the unperturbed state does not hold. With λ 100 m, the inverse wave number is k−1 15 m, so that h0 = 100 m already represents a deep ocean; in comparison, the typical scale on which non-uniformities in the mass density are relevant is rather 1 km. (aa) G. B Airy, 1801–1892 82 Waves in non-relativistic perfect fluids • For kh0  1, i.e in the case of a shallow sea with h0  λ, the approximation tanh(kh0 ) kh0 leads to the dispersion relation ω 2 = gh0 k 2 , i.e to phase and group velocities p cϕ = cg = gh0 , independent from the wavelength λ, signaling the absence of dispersive behavior. This phase velocity decreases with decreasing water

depth h0 . Accordingly, this might lead to an accumulation, similar to the case of a shock wave in Sec. V2, whose description however requires that one take into account the nonlinear terms in the equations, which have been discarded here. In particular, we have explicitly assumed |δh(t, x)|  h0 , in order to linearize the problem, so that considering the limiting case h0 0 is questionable. In addition, a temptation when investigating the small-depth behavior h0 0 is clearly to describe the breaking of waves as they come to shore. Yet the harmonic ansatz (V23) assumes that the Laplace equation is considered on a horizontally unbounded domain, i.e far from any coast, so again the dispersion relation (V.26) may actually no longer be valid The boundary condition (V.25b) provides us directly with the shape of the free surface of the sea, namely δh(t, x) = 1 ∂ϕ(t, x, z) g ∂t = z=h0 ωC cosh(kh0 ) sin(kx − ωt) ≡ δh0 sin(kx − ωt), g with δh0 ≡ (ωC/g) cosh(kh0 ) the

amplitude of the wave, which must remain much smaller than h0 . The profile of the surface waves of Airy’s linear theoryor rather its cross sectionis thus a simple sinusoidal curve. This shape automatically suggests a generalization, which is a first step towards taking into account nonlinearities, such that the free surface profile is sum of (a few) harmonics sin(kx−ωt), sin 2(kx − ωt), sin 3(kx − ωt). The approach leading to such a systematically expanded profile, which relies on a perturbative expansion to deal with the (still small) nonlinearities, is that of the Stokes waves. The gradient of the potential (V.23) yields (the components of) the flow velocity kg cosh(kz) δh0 sin(kx − ωt), ω cosh(kh0 ) kg sinh(kz) vz (t, x, z) = − δh0 cos(kx − ωt). ω cosh(kh0 ) vx (t, x, z) = Integrating these functions with respect to time leads to the two functions kg δh0 cosh(kz) δh0 cosh(kz) cos(kx − ωt) = x0 + cos(kx − ωt), ω 2 cosh(kh0 ) sinh(kh0 ) δh0

sinh(kz) kg δh0 sinh(kz) sin(kx − ωt) = z0 + sin(kx − ωt), z(t) = z0 + 2 ω cosh(kh0 ) sinh(kh0 ) x(t) = x0 + with x0 and z0 two integration constants. Choosing x0 x and z0 z, if δh0  k −1 , these functions represent the components of the trajectory (pathline) of a fluid particle that is at time t in the vicinity of the point with coordinates (x, z), and whose velocity at that time is thus approximately the flow velocity ~v(t, x, z). Since  2  2 [x(t) − x0 ]2 [z(t) − z0 ]2 kg δh0 δh0 + = = , ω 2 cosh(kh0 ) sinh(kh0 ) cosh2 (kz) sinh2 (kz) this trajectory is an ellipse, whose major and minor axes decrease with increasing depth h0 − z. In the deep sea case kh0  1, one can use the approximations sinh(kz) cosh(kz) ekz /2 for 1  kz . kh0 , which shows that the pathlines close to the sea surface are approximately circles 83 V.3 Gravity waves Eventually, the pressure distribution in the sea follows from Eq. (V24) in which one uses the velocity potential

(V.23), resulting in   ∂ϕ(t, x, z) cosh(kz) P (t, x, z) = P 0 + ρg(h0 − z) + ρ = P 0 + ρg h0 − z + δh0 sin(kx − ωt) . ∂t cosh(kh0 ) The contribution P 0 + ρg(h0 − z) is the usual hydrostatic one, corresponding to the unperturbed sea, while the effect of the surface wave is proportional to its amplitude δh0 and decreases with increasing depth. V.32 Solitary waves We now want to go beyond the linear limit considered in § V.31 c for waves at the free surface of a liquid in a gravity field. To that extent, we shall take a few steps back, and first rewrite the dynamical equations of motion and the associated boundary conditions in a dimensionless form (§ V.32 a) This formulation involves two independent parameters, and we shall focus on the limiting case where both are smallyet non-vanishingand obey a given parametric relation. In that situation, the equation governing the shape of the free surface is the Korteweg–de Vries equation, which in particular describes

solitary waves (§ V.32 c)(28) V.32 a Dimensionless form of the equations of motion ::::::::::::::::::::::::::::::::::::::::::::::::::::::: As in § V.31 c, the equations governing the dynamics of gravity waves at the surface of the sea are on the one hand the incompressibility condition ~ ·~v(t,~r) = 0, ∇ and on the other hand the Euler equation  ∂~v(t,~r)  ~ ~v(t,~r) = − 1 ∇ ~ P (t,~r) − g~ez . + ~v(t,~r) · ∇ ∂t ρ (V.27a) (V.27b) The boundary conditions (V.21) they obey are the absence of vertical velocity at the sea bottom vz (t, x, z = 0) = 0, (V.27c) the identity of the sea vertical velocity with the rate of change of the surface altitude h0 + δh(t, x)
∂δh(t, x) ∂δh(t, x) vz t, x, z = h0 +δh(t, x) = + vx (t,~r) , (V.27d) ∂t ∂x and finally the existence of a uniform pressure at that free surface
P t, x, z = h0 +δh(t, x) = P 0 . In the sea at rest, the pressure field is given by the hydrostatic formula P st.(t, x, z) = P 0 − ρg(h0 − z)

Defining the “dynamical pressure” in the sea water as P dyn. ≡ P − P st , one finds first that the right ~ P dyn. , and secondly that the hand side of the Euler equation (V.27b) can be replaced by −(1/ρ)∇ boundary condition at the free surface becomes
(V.27e) P dyn. t, x, z = h0 +δh(t, x) = ρgδh(t, x) Let us now recast Eqs. (V27) in a dimensionless form For that extent, we introduce two characteristic lengths: Lc for long-wavelength motions along x or z, and δhc for the amplitude of the surface deformation; for durations, we define a scale tc , which will later be related to Lc with the help of a typical velocity. With these scales, we can construct dimensionless variables t x z t∗ ≡ , x∗ ≡ , z∗ ≡ , tc Lc Lc (28) This Section follows closely the Appendix A of Ref. [18] 84 Waves in non-relativistic perfect fluids and fields: δh∗ ≡ δh , δhc vx∗ ≡ vx , δhc /tc vz∗ ≡ vz , δhc /tc P∗ ≡ P dyn. ρ δhc Lc /t2c . Considering

the latter as functions of the reduced variables t∗ , x∗ , z ∗ , one can rewrite the equations (V.27a)–(V27e) The incompressibility thus becomes ∂ vx∗ ∂ vz∗ + = 0, ∂x∗ ∂z ∗ and the Euler equation, projected successively on the x and z directions   ∗ ∗ ∂P ∗ ∂ vx∗ ∗ ∂ vx ∗ ∂ vx = − + ε v + v , x z ∂t∗ ∂x∗ ∂z ∗ ∂x∗ (V.28a) (V.28b) and   ∗ ∗ ∂P ∗ ∂ vz∗ ∗ ∂ vz ∗ ∂ vz = − + ε v + v . (V.28c) x z ∂t∗ ∂x∗ ∂z ∗ ∂z ∗ where we have introduced the dimensionless parameter ε ≡ δhc /Lc . In turn, the various boundary conditions are (V.28d) vz∗ = 0 at z ∗ = 0 at the sea bottom, and at the free surface vz∗ = ∗ ∂δh∗ ∗ ∂δh + εv x ∂t∗ ∂x∗ at z ∗ = δ + εδh∗ (V.28e) with δ ≡ h0 /Lc , and P∗ = gt2c ∗ δh Lc at z ∗ = δ + εδh∗ . Introducing the further dimensionless number p Fr ≡ Lc /g tc the latter condition becomes P∗ = 1 ∗ 2 δh

Fr at z ∗ = δ + εδh∗ (V.28f) Inspecting these equations, one sees that the parameter ε controls the size of nonlinearitiescf. Eqs. (V28b), (V28c) and (V28e), while δ measures the depth of the sea in comparison to the typical wavelength Lc . Both parameters are a priori independent: δ is given by the physical setup we want to describe, while ε quantifies the amount of nonlinearity we include in the description. To make progress, we shall from now on focus on gravity waves on shallow water, i.e assume δ  1. In addition, we shall only consider small nonlinearities, ε  1 To write down expansions in a consistent manner, we shall assume that the two small parameters are not of the same order, but rather that they obey ε ∼ δ 2 . Calculations will be considered up to order O(δ 3 ) or equivalently O(δε). For the sake of brevity, we now drop the subscript ∗ from the dimensionless variables and fields. V.32 b Velocity potential ::::::::::::::::::::::::: If the flow

is irrotational, ∂ vx /∂z = ∂ vz /∂x, so that one may transform Eq. (V28b) into   ∂ vx ∂ vx ∂ vz 1 ∂δh + ε vx + vz + 2 = 0. (V.29) ∂t ∂x ∂x Fr ∂x ~ In addition, one may introduce a velocity potential ϕ(t, x, z) such that ~v = −∇ϕ. With the latter, 85 V.3 Gravity waves the incompressibility condition (V.28a) becomes the Laplace equation ∂2ϕ ∂2ϕ + 2 = 0. ∂x2 ∂z The solution for the velocity potential will be written as an infinite series in z ϕ(t, x, z) = ∞ X z n ϕn (t, x), (V.30) (V.31) n=0 with unknown functions ϕn (t, x). Substituting this ansatz in the Laplace equation (V30) gives after some straightforward algebra  2  ∞ X n ∂ ϕn (t, x) + (n + 1)(n + 2)ϕn+2 (t, x) = 0. z ∂x2 n=0 In order for this identity to hold for arbitrary zat least, for the values relevant for the flow, each coefficient should individually vanish, i.e the ϕn should obey the recursion relation ϕn+2 (t, x) = − ∂ 2 ϕn (t, x) 1 (n +

1)(n + 2) ∂x2 for n ∈ N. (V.32) It is thus only necessary to determine ϕ0 and ϕ1 to know the whole series. The boundary condition (V.28d) at the bottom reads ∂ϕ(t, x, z = 0)/∂z = 0 for all t and x, which implies ϕ1 (t, x) = 0, so that all ϕ2n+1 identically vanish. As a consequence, ansatz (V31) with the recursion relation (V.32) give ϕ(t, x, z) = ϕ0 (t, x) − z 2 ∂ 2 ϕ0 (t, x) z 4 ∂ 4 ϕ0 (t, x) + + . 2 ∂x2 4! ∂x4 ~ Differentiating with respect to x or z yields the components of the velocity ~v = −∇ϕ ∂ϕ0 (t, x) z 2 ∂ 3 ϕ0 (t, x) z 4 ∂ 5 ϕ0 (t, x) + − + . ∂x 2 ∂x3 4! ∂x5 ∂ 2 ϕ0 (t, x) z 3 ∂ 4 ϕ0 (t, x) vz (t, x, z) = z − + . ∂x2 3! ∂x4 vx (t, x, z) = − Introducing the notation u(t, x) ≡ −∂ϕ0 (t, x)/∂x and anticipating that the maximal value of z relevant for the problem is of order δ, these components may be expressed as z 2 ∂ 2 u(t, x) + o (δ 3 ), 2 ∂x2 ∂u(t, x) z 3 ∂ 3 u(t, x) vz (t, x, z) = −z +

+ o (δ 3 ), ∂x 3! ∂x3 vx (t, x, z) = u(t, x) − (V.33a) (V.33b) where the omitted terms are beyond O(δ 3 ). Linear waves rediscovered If we momentarily set ε = 0which amounts to linearizing the equations of motion and boundary conditions, consistency requires that we consider equations up to order δ at most. That is, we keep only the first terms from Eqs. (V33): at the surface at z δ, they become vx (t, x, z = δ) u(t, x), vz (t, x, z = δ) −δ ∂u(t, x) , ∂x (V.34a) while the boundary condition (V.28e) simplifies to ∂δh(t, x) ∂φ(t, x) =δ , ∂t ∂t where we have introduced φ(t, x) ≡ δh(t, x)/δ. vz (t, x, z = δ) = (V.34b) 86 Waves in non-relativistic perfect fluids Meanwhile, Eq. (V29) with ε = 0 reads ∂ vx (t, x) δ ∂φ(t, x) + 2 = 0. ∂t ∂x Fr (V.34c) Together, Eqs. (V34a)–(V34c) yield after some straightforward manipulations the equation δ ∂ 2 u(t, x) ∂ 2 u(t, x) − =0 ∂t2 Fr2 ∂x2 (V.35) √ √ i.e a linear

equation describing waves with the dimensionless phase velocity δ/Fr = gh0 /(Lc /tc ) Since √ the scaling factor of x resp. t is Lc resp tc , the corresponding dimensionful phase velocity is cϕ = gh0 , as was already found in § V.31 c for waves on shallow sea √ Until now, the scaling factor tc was independent from Lc . Choosing tc ≡ Lc / gh0 , ie the unit in which times are measured, the factor δ/Fr2 equals 1, leading to the simpler-looking equation   ∂φ(t, x) ∂ vx (t, x, z) ∂ vx (t, x, z) ∂ vz (t, x, z) + + ε vx (t, x, z) + vz (t, x, z) =0 (V.36) ∂t ∂x ∂x ∂x instead of Eq. (V29) V.32 c Non-linear waves on shallow water ::::::::::::::::::::::::::::::::::::::::::: Taking now ε 6= 0 and investigating the equations up to order O(δ 3 ), O(δε), Eqs. (V33) at the free surface at z = δ(1 + εφ) become
δ 2 ∂ 2 u(t, x) , vx t, x, z = δ(1 + εφ) = u(t, x) − 2 ∂x2 (V.37a)
  ∂u(t, x) δ 3 ∂ 3 u(t, x) vz t, x, z = δ(1 + εφ) = −δ 1 +

εφ(t, x) + . ∂x 6 ∂x3 (V.37b) Inserting these velocity components in (V.36) while retaining only the relevant orders yields ∂u(t, x) ∂φ(t, x) ∂u(t, x) δ 2 ∂ 3 u(t, x) − + εu(t, x) + = 0. 2 ∂t 2 ∂t ∂x ∂x ∂x (V.38) On the other hand, the velocity components are also related by the boundary condition (V.28e), which reads

∂φ(t, x) ∂φ(t, x) vz t, x, z = δ(1 + εφ) = δ + δεvx t, x, z = δ(1 + εφ) . ∂t ∂x Substituting Eq. (V37a) resp (V37b) in the right resp left member yields  ∂u(t, x) δ 2 ∂ 3 u(t, x) ∂φ(t, x)  ∂φ(t, x) + εu(t, x) + 1 + εφ(t, x) − = 0. ∂t ∂x ∂x 6 ∂x3 (V.39) To leading order in δ and ε, the system of nonlinear partial differential equations (V.38)–(V39) simplifies to the linear system  ∂u(t, x) ∂φ(t, x)   + =0  ∂t ∂x    ∂φ(t, x) + ∂u(t, x) = 0, ∂t ∂x which admits the solution u(t, x) = φ(t, x) under the condition ∂u(t, x) ∂u(t, x) + = 0, ∂t ∂x

(V.40) which describes a traveling wave with (dimensionless) velocity 1, u(t, x) = u(x−t). We again recover the linear sea surface waves which we have already encountered twice. 87 V.3 Gravity waves Going to next-to-leading order O(δ 2 ), O(ε), we look for solutions in the form u(t, x) = φ(t, x) + εu(ε)(t, x) + δ 2 u(δ)(t, x) (V.41) with φ, u(ε) , u(δ) functions that obey condition (V.40) up to terms of order ε or δ 2 Inserting this ansatz in Eqs. (V38)–(V39) yields the system  (ε) 2 3 (δ)   ∂φ + ∂φ + ε ∂u + δ 2 ∂u + 2εφ ∂φ − δ ∂ φ = 0  ∂t ∂x ∂x ∂x ∂x 6 ∂x3 3 (ε) 2 (δ)    ∂φ + ∂φ + ε ∂u + δ 2 ∂u + εφ ∂φ − δ ∂ φ = 0, ∂t ∂x ∂t ∂t ∂x 2 ∂x2 ∂t where for the sake of brevity, the (t, x)-dependence of the functions was not written. Subtracting both equations and using condition (V.40) to relate the time and space derivatives of φ, u(ε) , and u(δ) , one finds  (δ)  

 (ε) ∂φ(t, x) 1 ∂ 3 φ(t, x) ∂u (t, x) 1 2 ∂u (t, x) +δ = 0. ε + φ(t, x) − ∂x 2 ∂x ∂x 3 ∂x3 Since the two small parameters ε and δ are independent, each term between square brackets in this identity must identically vanish. Straightforward integrations then yield 1 u(ε)(t, x) = − φ(t, x) + C (ε)(t), 4 u(δ)(t, x) = 1 ∂ 2 φ(t, x) + C (δ)(t), 3 ∂x2 with C (ε) , C (δ) two functions of time only. These functions can then be substituted in the ansatz (V.41) Inserting the latter in Eq (V39) yields an equation involving the unknown function φ only, namely ∂φ(t, x) 1 2 ∂ 3 φ(t, x) ∂φ(t, x) ∂φ(t, x) 3 + + εφ(t, x) + δ = 0. ∂t ∂x 2 ∂x 6 ∂x3 (V.42) The first two terms only are those of the linear-wave equation of motion (V.40) Since the ε and δ nonlinear corrections also obey the same condition, it is fruitful to perform a change of variables from (t, x) to (τ, ξ) with τ ≡ t, ξ ≡ x − t. Equation (V42) then becomes

∂φ(τ, ξ) 1 2 ∂ 3 φ(τ, ξ) ∂φ(τ, ξ) 3 + εφ(τ, ξ) + δ = 0, ∂τ 2 ∂ξ 6 ∂ξ 3 (V.43) which is the Korteweg–de Vries equation.(ab),(ac) Remark: By rescaling the variables τ and ξ to a new set (τ, ξ), one can actually absorb the parameters ε, δ which were introduced in the derivation. Accordingly, the more standard form of the Korteweg–de Vries (KdV) equation is ∂φ(τ, ξ) ∂ 3 φ(τ, ξ) ∂φ(τ, ξ) + 6φ(τ, ξ) + = 0. ∂τ ∂ξ ∂ξ3 (V.44) Solitary waves The Korteweg–de Vries equation admits many different solutions. Among those, there is the class of solitary waves or solitons, which describe signals that propagate without changing their shape. (ab) D. Korteweg, 1848–1941 (ac) G. de Vries, 1866–1934 88 Waves in non-relativistic perfect fluids A specific subclass of solitons of the KdV equation of special interest in fluid dynamics consists of those which at each given instant vanish at (spatial) infinity. As solutions of

the normalized equation (V.44), they read φ0 (V.45a) φ(τ, ξ) = p  2 cosh φ0 /2 (ξ − 2φ0 τ) with φ0 the amplitude of the wave. Note that φ0 must be nonnegative, which means that these solutions describe bumps above the mean sea levelwhich is the only instance of such solitary wave observed experimentally. Going back first to the variables (τ, ξ), then to the dimensionless variables (t∗ , x∗ ), and eventually to the dimensionful variables (t, x) and field δh, the soliton solution reads δhmax r δh(t, x) = (V.45b)      , √ 1 3δhmax δhmax 2 cosh x − gh0 1+ t 2h0 h0 2h0 with δhmax the maximum amplitude of the solitary wave. This solution, represented in Fig V2, has a few properties that can be read directly off its expression and differ from those of linear sea surface waves, namely • the propagation velocity csoliton of the solitonwhich is the factor in front of tis larger than for linear waves; • the velocity csoliton increases with the amplitude

δhmax of the soliton; • the width of the soliton decreases with its amplitude. δh(t, x) δhmax = 1, t = t0 δhmax = 0.25, t = t0 δhmax = 1, t = t1 > t0 δhmax = 0.25, t = t1 x Figure V.2 – Profile of the soliton solution (V45) Bibliography for Chapter V • National Committee for Fluid Mechanics film & film notes on Waves in Fluids; • Guyon et al. [2] Chapter 64; • Landau–Lifshitz [3, 4] Chapters I § 12, VIII § 64–65, IX § 84–85, and X § 99; • Sommerfeld [5, 6] Chapters III § 13, V § 23, 24 & 26 and VII § 37. C HAPTER VI Non-relativistic dissipative flows The dynamics of Newtonian fluids is entirely governed by a relatively simple set of equations, namely the continuity equation (III.9), the Navier–Stokes equation (III31), andwhen phenomena related with temperature gradients become relevantthe energy conservation equation (III.35) As in the case of perfect fluids, there are a priori more unknown dynamical fields than equations, so that an

additional relation has to be provided, either a kinematic constraint or an equation of state. In this Chapter and the next two ones, a number of simple solutions of these equations are presented, together with big classes of phenomena that are accounted in various more or less simplified situations. With the exception of the static-fluid case, in which the only novelty with respect to the hydrostatics of perfect fluids is precisely the possible transport of energy by heat conduction (Sec. VI11), the motions of interest in the present Chapter are mostly laminar flows in which viscous effects play an important role while heat transport is negligible. Thus, the role of the no-slip condition at a boundary of the fluid is illustrated with a few chosen examples of stationary motions within idealized geometrical setups (Sec. VI1) By introducing flow-specific characteristic length and velocity scales, the Navier–Stokes equation can be rewritten in a form involving only dimensionless

variables and fields, together with parameterslike for instance the Reynolds number. These parameters quantify the relative importance of the several physical effects likely to play a role in a motion (Sec VI2) According to the value of the dimensionless numbers entering the dynamical equations, the latter may possibly be simplified. This leads to simpler equations with limited domain of validity, yet which become more easily tractable, as exemplified by the case of flows in which shear viscous effects predominate over the influence of inertia (Sec. VI3) Another simplified set of equations can be derived to describe the fluid motion in the thin layer close to a boundary of the flow, in which the influence of this boundary plays a significant role (Sec VI.4) Eventually, the viscosity-induced modifications to the dynamics of vorticity (Sec. VI5) and to the propagation of sound waves (Sec. VI6) are presented VI.1 Statics and steady laminar flows of a Newtonian fluid In this Section, we

first write down the equations governing the statics of a Newtonian fluid (Sec. VI11), then we investigate a few idealized stationary laminar fluid motions, in which the velocity field is entirely driven by the no-slip condition at boundaries (Secs. VI12–VI14) VI.11 Static Newtonian fluid Consider a motionless [~v(t,~r) = ~0] Newtonian fluid in an external gravitational potential Φ(~r) or more generally, submitted to conservative volume forces such that ~ f~V (t,~r) = −ρ(t,~r) ∇Φ(t,~ r). (IV.1) 90 Non-relativistic dissipative flows The three coupled equations (III.9), (III31) and (III35) respectively simplify to ∂ρ(t,~r) = 0, ∂t (VI.1a) from where follows the time independence of the mass density ρ(t,~r), ~ P (t,~r) = −ρ(t,~r) ∇Φ(t,~ ~ ∇ r), (VI.1b) similar to the fundamental equation (IV.2) governing the hydrostatics of a perfect fluid, and  ∂e(t,~r) ~  ~ (t,~r) , = ∇ · κ(t,~r)∇T (VI.1c) ∂t which describes the transport of energy without

macroscopic fluid motion, i.e non-convectively, thanks to heat conduction. VI.12 Plane Couette flow In the example of this Section and the next two ones (Secs. VI13–VI14), we consider steady, incompressible, laminar flows, in absence of significant volume forces. Since the mass density ρ is fixed, thus known, only four equations are needed to determine the flow velocity ~v(~r) and pressure P (~r), the simplest possibility being to use the continuity and Navier–Stokes equations. In the stationary and incompressible regime, these become ~ ·~v(~r) = 0 ∇ (VI.2a)   ~ P (~r) + ν4~v(~r), ~ ~v(~r) = − 1 ∇ ~v(~r) · ∇ ρ (VI.2b) with ν the kinematic shear viscosity, assumed to be the same throughout the fluid. The so-called (plane) Couette flow(ad) is, in its idealized version, the motion of a viscous fluid between two infinitely extended plane plates, as represented in Fig. VI1, where the lower plate is at rest, while the upper one moves in its own plane with a constant

velocity ~u. It will be assumed h 6 - ~v(y) - ~u- ? y 6 - x Figure VI.1 – Setup of the plane Couette flow that the same pressure P ∞ holds ”at infinity”, in any direction. As the flow is assumed to be laminar, the geometry of the problem is invariant under arbitrary translations in the (x, z)-plane. This is automatically taken into account by the ansatz~v(~r) = v(y)~ex for the flow velocity. Inserting this form in Eqs (VI2) yields ∂ v(y) = 0, ∂x v(y) ∂ v(y) d2 v(y) 1~ P (~r) + ν ~ex = − ∇ ~ex . ∂x ρ dy 2 (VI.3a) (VI.3b) With the ansatz for ~v(~r), the first equation is automatically fulfilled, while the term on the left hand side of the second equation vanishes. Projecting the latter on the y and z directions thus yields ∂ P (~r)/∂y = 0expressing the assumed absence of sizable effects from gravityand (ad) M. Couette, 1858–1943 VI.1 Statics and steady laminar flows of a Newtonian fluid 91 ∂ P (~r)/∂z = 0since the problem is independent

of z. Along the x direction, one finds ∂ P (~r) d2 v(y) =η . ∂x dy 2 (VI.4) Since the right member of this equation is independent of x and z, a straightforward integration gives P (~r) = α(y)x + β(y), where the functions α, β only depend on y. These functions are determined by the boundary conditions: since P (x = −∞) = P (x = ∞) = P ∞ , then α(y) = 0, β(y) = P ∞ , and Eq. (VI4) simplifies to d2 v(y) = 0. dy 2 This yields v(y) = γy + δ, with γ and δ two integration constants, which are again fixed by the boundary conditions. At each plate, the relative velocity of the fluid with respect to the plate must vanish: v(y = 0) = 0, v(y = h) = |~u|, leading to δ = 0 and γ = |~u|/h. All in all, the velocity thus depends linearly on y ~v(~r) = y ~u for 0 ≤ y ≤ h. h Consider now a surface element d2 S. The contact force d2 F~s exerted on it by the fluid follows from the Cauchy stress tensor, whose Cartesian components (III.27c) here read   |~u|   i −

P η 0 h ∂ v (~r) ∂ vj (~r) ∼  |~u|∞  + σ ij (~r) = −P (~r)δ ij + η = ηh −P ∞ 0 . ∂xj ∂xi 0 0 −P ∞ The force per unit surface on the motionless plate at y = 0, corresponding to a unit normal vector ~en (~r) = ~ey , is  |~u|   X 3 3 ηh 2 X ~
d Fs (~r) ij ij ~s (~r) =  ~ e = · ~ e = σ (~ r) ~ e · ~ e = T σ (~ r)~ e ⊗ ~ e −P ∞  . y j y i i j d2 S i,j=1 i,j=1 0 Due to the friction exerted by the fluid, the lower plate is dragged by the flow in the (positive) x direction. Remark: The tangential stress on the lower plate is η~u/h, proportional to the shear viscosity: measuring the tangential stress with known |~u| and h yields a measurement of η. In practice, this measurement rather involves the more realistic cylindrical analog to the above plane flow, the so-called Couette–Taylor flow .(ae) VI.13 Plane Poiseuille flow Let us now consider the flow of a Newtonian fluid between two motionless plane plates with a finite

length along the x directionyet still infinitely extended along the z direction, as illustrated in Fig. VI2 The pressure is assumed to be different at both ends of the plates in the x direction, amounting to the presence of a pressure gradient along x. Assuming for the flow velocity ~v(~r) the same form v(y)~ex , independent of x, as in the case of the plane Couette flow, the equations of motion governing v(y) and pressure P (~r) are the same as in the previous Section VI.12, namely Eqs (VI3)–(VI4) The boundary conditions are however different. (ae) G. I Taylor, 1886–1975 92 Non-relativistic dissipative flows P1 6 -- h ?  P2 y -- 6 - x - L Figure VI.2 – Flow between two motionless plates for P 1 > P 2 , i.e ∆P > 0 Thus, P 1 6= P 2 results in a finite constant pressure gradient along x, α = ∂ P (~r)/∂x = −∆P /L 6= 0, with ∆P ≡ P 1 − P 2 the pressure drop. Equation (VI4) then leads to v(y) = − 1 ∆P 2 y + γy + δ, 2η L with γ and

δ two new constants. The “no-slip” boundary conditions for the velocity at the two plates read v(y = 0) = 0, v(y = h) = 0, 1 ∆P h. The flow velocity thus has the parabolic profile 2η L  1 ∆P  y(h − y) for 0 ≤ y ≤ h, v(y) = 2η L which leads to δ = 0 and γ = (VI.5) directed along the direction of the pressure gradient. Remark: The flow velocity (VI.5) becomes clearly problematic in the limit η 0! Tracing the problem back to its source, the equations of motion (VI.3) cannot hold with a finite gradient along the x direction and a vanishing viscosity. One quickly checks that the only possibility in the case of a perfect fluid is to drop one of the assumptions, either incompressibility or laminarity. VI.14 Hagen–Poiseuille flow The previous two examples involved plates with an infinite length in at least one direction, thus were idealized constructions. In contrast, an experimentally realizable fluid motion is that of the Hagen–Poiseuille flow ,(af) in which

a Newtonian fluid flows under the influence of a pressure gradient in a cylindrical tube with finite length L and radius a (Fig. VI3) Again, the motion is assumed to be steady, incompressible and laminar. a P 1   P 2 -z - L Figure VI.3 – Setup of the Hagen–Poiseuille flow p Using cylindrical coordinates, the ansatz ~v(~r) = v(r)~ez with r = x2 + y 2 satisfies the conti~ ·~v(~r) = 0 and gives for the incompressible Navier–Stokes equation nuity equation ∇  ∂ P (~r) ∂ P (~r)    ∂x = ∂y = 0 ~ P (~r) = η4~v(~r) ⇔  2   2  ∇ (VI.6) ∂ v(r) ∂ 2 v(r)) ∂ P (~r) d v(r) 1 dv(r)    =η + = η + . ∂z ∂x2 ∂y 2 dr2 r dr (af) G. Hagen, 1797–1884 93 VI.1 Statics and steady laminar flows of a Newtonian fluid The right member of the equation in the second line is independent of z, implying that the pressure gradient along the z direction is constant: ∂ P (~r) ∆P =− , ∂z L with ∆P ≡ P 1 − P 2 . The z component of the

Navier–Stokes equation (VI6) thus becomes ∆P d2 v(r) 1 dv =− . + dr2 r dr ηL (VI.7) As always, this linear differential equation is solved in two successive steps, starting with the associated homogeneous equation. To find the general solution of the latter, one may introduce χ(r) ≡ dv(r)/dr, which satisfies the simpler equation dχ(r) χ(r) + = 0. dr r The generic solution is ln χ(r) = − ln r + const., ie χ(r) = A/r with A a constant This then leads to v(r) = A ln r + B with B an additional constant. A particular solution of the inhomogeneous equation (VI.7) is v(r) = Cr2 with C = −∆P /4ηL The general solution of Eq. (VI7) is then given by ∆P 2 r , 4ηL where the two integration constants still need to be determined. To have a regular flow velocity at r = 0, the constant A should vanish. In turn, the boundary condition at the tube wall, v(r = a) = 0, determines the value of the constant B = (∆P /4ηL)a2 . All in all, the velocity profile thus reads
∆P 2

v(r) = a − r2 for r ≤ a. (VI.8) 4ηL v(r) = A ln r + B − This is again parabolic, with ~v in the same direction as the pressure drop. The mass flow rate across the tube cross section follows from a straightforward integration: Z a Z
∆P a 2 ∆P a4 πρa4 ∆P Q= ρv(r) 2πr dr = 2πρ a r − r3 dr = 2πρ = . (VI.9) 4ηL 0 4ηL 4 8η L 0 This result is known as Hagen–Poiseuille law (or equation) and means that the mass flow rate is proportional to the pressure drop per unit length. Remarks: ∗ The Hagen–Poiseuille law only holds under the assumption that the flow velocity vanishes at the tube walls. The experimental confirmation of the lawwhich was actually deduced from experiment by Hagen (1839) and Poiseuille (1840)is thus a proof of the validity of the no-slip assumption for the boundary condition. ∗ The mass flow rate across the tube cross section may be used to define that average flow velocity as Q = πa2 ρhvi with Z a 1 1 hvi ≡ v(r) 2πr dr = v(r = 0).

πa2 0 2 The Hagen–Poiseuille law then expresses a proportionality between the pressure drop per unit length and hvi in a laminar flow. Viewing ∆P /L as the “generalized force” driving the motion, the corresponding “response” hvi of the fluid is thus linear. The relation is quite different in the case of a turbulent flow with the same geometry: for instance, measurements by Reynolds [19] gave ∆P /L ∝ hvi1.722 94 Non-relativistic dissipative flows VI.2 Dynamical similarity ~ v(t,~r) = 0 The incompressible motion of a Newtonian fluid is governed by the continuity equation ∇·~ and the Navier–Stokes equation (III.32) In order to determine the relative influence of the various terms of the latter, it is often convenient to consider dimensionless forms of the incompressible Navier–Stokes equation, which leads to the introduction of a variety of dimensionless numbers. For instance, the effect of the fluid mass density ρ and shear viscosity η (or equivalently

ν), which are uniform throughout the fluid, on a flow in the absence of volume forces is entirely encoded in the Reynolds number (Sec. VI21) Allowing for volume forces, either due to gravity or to inertial forces, their relative influence is controlled by similar dimensionless parameters (Sec. VI22) Let Lc resp. vc be a characteristic length resp velocity for a given flow Since the Navier–Stokes equation itself does not involve any parameter with the dimension of a length or a velocity, both are controlled by “geometry”, by the boundary conditions for the specific problem under consideration. Thus, Lc may be the size (diameter, side length) of a tube in which the fluid flows or of an obstacle around which the fluid moves. In turn, vc may be the uniform velocity far from such an obstacle With the help of Lc and vc , one may rescale the physical quantities in the problem, so as to obtain dimensionless quantities, which will hereafter be denoted with ∗ : ~r∗ ≡ ~r , Lc ~v∗

≡ ~v , vc t∗ ≡ t , Lc /vc P∗ ≡ P − P0 ρvc2 , (VI.10) where P 0 is some characteristic value of the (unscaled) pressure. VI.21 Reynolds number Consider first the incompressible Navier–Stokes equation in the absence of external volume forces. Rewriting it in terms of the dimensionless variables and fields (VI10) yields ∂~v∗(t∗,~r∗ )  ∗ ∗ ∗ ~ ∗  ∗ ∗ ∗ ~ ∗P ∗(t∗,~r∗ ) + η 4∗~v∗(t∗,~r∗ ), + ~v (t ,~r ) · ∇ ~v (t ,~r ) = −∇ ∗ ∂t ρvc Lc (VI.11) ~ ∗ resp. 4∗ the gradient resp Laplacian with respect to the reduced position variable ~r∗ mit ∇ Besides the reduced variables and fields, this equation involves a single dimensionless parameter, the Reynolds number Re ≡ ρvc Lc vc Lc = . η ν (VI.12) This number measures the relative importance of inertia and viscous friction forces on a fluid element or a body immersed in the moving fluid: at large resp. small Re, viscous effects are negligible resp

predominant. Remark: As stated above Eq. (VI10), both Lc and vc are controlled by the geometry and boundary conditions. The Reynolds numberand every similar dimensionless we shall introduce hereafteris thus a characteristic of a given flow, not of the fluid. Law of similitude(lxii) ::::::::::::::::: The solutions for the dynamical fields ~v∗ , P ∗ at fixed boundary conditions and geometry specified in terms of dimensionless ratios of geometrical lengthsare functions of the independent variables t∗ , ~r∗ , and of the Reynolds number: ~v∗(t∗,~r∗ ) = ~f1 (t∗,~r∗, Re), P ∗(t∗,~r∗) = f2 (t∗,~r∗, Re), with ~f1 resp. f2 a vector resp scalar function Flow velocity and pressure are then given by (lxii) Ähnlichkeitsgesetz (VI.13) 95 VI.2 Dynamical similarity   vc t ~r ~ ~v(t,~r) = vc f1 , , Re , Lc Lc P (t,~r) = P 0 + ρvc2 f2   vc t ~r , , Re . Lc Lc These equations underlie the use of fluid dynamical simulations with experimental models at a

reduced scale, yet possessing the same (rescaled) geometry. Let Lc , vc resp LM , vM be the characteristic lengths of the real-size flow resp of the reduced-scale experimental flow; for simplicity, we assume that the same fluid is used in both cases. If vM /vc = Lc /LM , the Reynolds number for the experimental model is the same as for the real-size fluid motion: both flows then admit the same solutions ~v∗ and P ∗ , and are said to be dynamically similar . Remark: The functional relationships between the “dependent variables” ~v∗ , P ∗ and the “independent variables” t∗ , ~r∗ and a dimensionless parameter (Re) represent a simple example of the more general (Vaschy(ag) –)Buckingham(ah) π-theorem [20] in dimensional analysis, see e.g Refs [21, 22] Chapter 7 or [23]. VI.22 Other dimensionless numbers If the fluid motion is likely to be influenced by gravity, the corresponding volume force density f~V = −ρ~g must be taken into account in the right member of the

incompressible Navier–Stokes equation (III.32) Accordingly, if the latter is written in dimensionless form as in the previous Section, there will come an additional term on the right hand side of Eq. (VI11), proportional to 1/Fr2 , with vc (VI.14) Fr ≡ √ gLc the Froude number .(ai) This dimensionless parameter measures the relative size of inertial and gravitational effects in the flow, the latter being important when Fr is small. In the presence of gravity, the dimensionless dynamical fields ~v∗ , P ∗ become functions of the reduced variables t∗ , ~r∗ controlled by both parameters Re and Fr. The Navier–Stokes equation (III.31) holds in an inertial frame In a non-inertial reference frame, there come additional terms, which may be expressed as fictive force densities on the right hand side, which come in addition to the “physical” volume force density f~V In the case of a reference frame in ~ 0 , there are thus two extra uniform rotation (with respect to an inertial

frame) with angular velocity Ω 
 ~ ~ −1 Ω ~ 0 × ~r 2 contributions corresponding to centrifugal and Coriolis forces, namely fcent. = −ρ∇ 2 ~ 0 ×~v, respectively. and f~Cor. = −2ρ Ω The relative importance of the latter in a given flow can be estimated with dimensionless numbers. Thus, the Ekman number (aj) ν η = (VI.15) Ek ≡ ρΩL2c ΩL2c measures the relative size of (shear) viscous and Coriolis forces, with the latter predominating over the former when Ek  1. One may also wish to compare the influences of the convective and Coriolis terms in the Navier– Stokes equation. This is done with the help of the Rossby number (ak) vc Ro ≡ (VI.16) ΩLc which is small when the effect of the Coriolis force is the dominant one. Remark: Quite obviously, the Reynolds (VI.12), Ekman (VI15), and Rossby (VI16) numbers obey the simple identity Ro = Re · Ek. (ag) (ak) A. Vaschy, 1857–1899 (ah) E Buckingham, 1867–1940 C.-G Rossby, 1898–1957 (ai) W. Froude,

1810–1879 (aj) V. Ekman, 1874–1954 96 Non-relativistic dissipative flows VI.3 Flows at small Reynolds number This Section deals with incompressible fluid motions at small Reynolds number Re  1, i.e in the situation in which shear viscous effects predominate over those of inertia in the Navier–Stokes equation. Such fluid motions are also referred to as Stokes flows or creeping flows(lxiii) VI.31 Physical relevance Equations of motion Flows of very different nature may exhibit a small Reynolds number (VI.12), because the latter combines physical quantities whose value can vary by many orders of magnitude in Nature.(29) A few examples of creeping flows are listed hereafter: • The motion of fluids past microscopic bodies; the small value of the Reynolds number then reflects the smallness of the length scale Lc ; for instance: – In water (η ≈ 10−3 Pa·s i.e ν ≈ 10−6 m2 · s−1 ), a bacteria of size Lc ≈ 5 µm “swims” with velocity vc ≈ 10 µm · s−1

, so that Re ≈ 5 · 10−5 for the motion of the water past the bacteria: if the bacteria stops propelling itself, the friction exerted by the water brings it immediately to rest.(30) Similarly, creeping flows are employed to describe the motion of reptiles in sandor more precisely, the flow of sand a past an undulating reptile [25]. – The motion of a fluid past a suspension of small size (Brownian) particles. This will be studied at further length in Sec. VI32 • The slow-velocity motion of geological material: in that case, the small value of vc and the large shear viscosity compensate the possibly large value of the typical length scale Lc . For example, the motion of the Earth’s mantle(31) with Lc ≈ 100 km, vc ≈ 10−5 m · s−1 , ρ ≈ 5 · 103 kg · m−3 and η ≈ 1022 Pa · s corresponds to a Reynolds number Re ≈ 5 · 10−19 . Note that the above examples all represent incompressible flows. For the sake of simplicity, we shall also only consider steady motions.

VI.31 a Stokes equation :::::::::::::::::::::::: Physically, a small Reynolds number means that the influence of inertia is much smaller than
~ that of shear viscosity. That is, the convective term ~v · ∇ ~v in the Navier–Stokes equation is negligible with respect to the viscous contribution. Assuming further stationaritywhich allows us to drop the time variableand incompressibility, the Navier–Stokes equation (III.31) simplifies to the Stokes equation ~ P (~r) = η4~v(~r) + f~V (~r). ∇ (VI.17) This constitutes a linearization of the incompressible Navier–Stokes equation. Using the relation     ~ × ∇ ~ × ~c(~r) = ∇ ~ ∇ ~ · ~c(~r) − 4~c(~r) ∇ (29) (VI.18) This is mostly true of the characteristic length and velocity scales and of the shear viscosity; in (non-relativistic) fluids, the mass density is always of the same order of magnitude, up to a factor 103 . (30) A longer discussion of the motion of bacteriafrom a physicist point of view, together with

the original formulation of the “scallop theorem”, can be found in Ref. [24] (31) From the mass density, the shear viscosity and the typical speed of sound cs ≈ 5000 m·s−1 of transverse waves i.e shear waves, that may propagate in a solid, but not in a fluid, one constructs a characteristic time scale tmantle = η/ρc2s ≈ 3000 years. For motions with a typical duration tc  tmantle , the Earth’s mantle behaves like a deformable solid: for instance, with respect to the propagation of sound waves following an earthquake. On the other hand, for motions on a “geological” time scale tc  tmantle , the mantle may be modeled as a fluid. (lxiii) schleichende Strömungen 97 VI.3 Flows at small Reynolds number valid for any vector field ~c(~r), the incompressibility condition, and the definition of vorticity, the Stokes equation can be rewritten as ~ P (~r) = −η ∇ ~ ×ω ∇ ~ (~r). (VI.19) As a result, the pressure satisfies the differential Laplace equation 4P

(~r) = 0. (VI.20) In practice, however, this equation is not the most useful, because the boundary conditions in a flow are mostly given in terms of the flow velocity, in particular at walls or obstacles, not of the pressure. Taking the curl of Eq. (VI19) and invoking again relation (VI18) remembering that the vorticity vector is itself already a curl, one finds 4~ ω (~r) = ~0, (VI.21) i.e the vorticity also obeys the Laplace equation We shall see in Sec VI5 that the more general dynamical equation obeyed by vorticity in Newtonian fluids does indeed yield Eq. (VI21) in the case of stationary, small Reynolds number flows. VI.31 b Properties of the solutions of the Stokes equation ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Thanks to the linearity of the Stokes equation (VI.17), its solutions possess various properties:(32) • Uniqueness of the solution at fixed boundary conditions. • Additivity of the solutions: if ~v1 and ~v2 are solutions of Eq. (VI17) with

respective boundary conditions, then the sum λ1~v1 + λ2~v2 with real numbers λ1 , λ2 is also a solution, for a problem with adequate boundary conditions. Physically, the multiplying factors should not be too large, to ensure that the Reynolds number of the new problem remains small. The multiplication of the velocity field ~v(~r) by a constant λ represents a change in the mass flow rate, while the streamlines (I.15) remain unchanged The dimensionless velocity field ~v∗ associated with the two solutions ~v(~r) and λ~v(~r) is the same, provided the differing characteristic velocities vc resp. λvc are used In turn, these define different values of the Reynolds number. For these solutions, ~v∗ as given by Eq (VI13) is
thus independent of the parameter Re, and thereby only depends on the variable ~r∗ : ~v = vc f~ ~r/Lc . This also holds for the corresponding dimensionless pressure P ∗ . Using dimensional arguments only, the tangential stress is η∂ vi /∂xj ∼ η vc /Lc

, so that the friction force on an object of linear size(33) Lc is proportional to η vc Lc . This result will now be illustrated on an explicit example [cf. Eq (VI26)], for which the computation can be performed analytically. VI.32 Stokes flow past a sphere Consider a sphere with radius R immersed in a fluid with mass density ρ and shear viscosity η, which far from the sphere flows with uniform velocity ~v∞ , as sketched in Fig. VI4 The goal is to determine the force exerted by the moving fluid on the sphere, which necessitates the calculation of the pressure and velocity in the flow. Given the geometry of the problem, a system of spherical coordinates (r, θ, ϕ) centered on the sphere center will be used. The Reynolds number Re = ρ|~v∞ |R/η is assumed to be small, so that the motion in the vicinity of the sphere can be modeled as a creeping flow, which is further taken to be incompressible. For the flow velocity, one looks for a stationary solution of the equations of

motion of the form (32) (33) Proofs can be found e.g in Ref [2, Chapter 823] As noted in the introduction to Sec. VI2, the characteristic length and velocity scales in a flow are precisely determined by the boundary conditions. 98 Non-relativistic dissipative flows ~eϕ ~v∞ ~er ϕ Figure VI.4 – Stokes flow past a sphere ~v(~r) = ~v∞ + ~u(~r), with the boundary condition ~u(~r) = ~0 for |~r| ∞. In addition, the usual impermeability and no-slip conditions hold at the surface of the sphere, resulting in the requirement ~u(|~r| = R) = −~v∞ . Using the linearity of the equations of motion for creeping flows, ~u obeys the equations   ~ ×~u(~r) = ~0 4 ∇ (VI.22a) and ~ ·~u(~r) = 0, ∇ (VI.22b) which comes from the incompressibility condition. ~ (~r). Using The latter equation is automatically satisfied if ~u(~r) is the curl of some vector field V dimensional considerations, the latter should depend linearly on the only explicit velocity scale in the problem,

namely ~v∞ . Accordingly, one makes the ansatz(34)   ~ (~r) = ∇ ~ × f (r)~v∞ = ∇f ~ (r) ×~v∞ , V with f (r) a function of r = |~r|, i.e f only depends on the distance from the sphere: apart from the direction of ~v∞ , which is already accounted for in the ansatz, there is no further preferred spatial direction, so that f should be spherically symmetric. ~ · [f (r)~v∞ ] = ∇f ~ (r) ·~v∞ then yield Relation (VI.18) together with the identity ∇   ~ (r) ·~v∞ − 4f (r)~v∞ . ~ ×V ~ (~r) = ∇ ~ ∇f ~u(~r) = ∇ (VI.23) The first term on the right hand side has a vanishing curl, and thus does not contribute when inserting ~u(~r) in equation (VI.22a):     ~ 4f (r) ×~v∞ , ~ ×~u(~r) = −∇ ~ × 4f (r)~v∞ = −∇ ∇ so that    ~ 4f (r) ×~v∞ = ~0. 4 ∇  
~ 4f (r) only has a comSince f (r) does not depend on the azimuthal and polar angles, 4 ∇ ponent along the radial direction 
with (unit) basis vector ~er ; as thus, it cannot be

always parallel ~ 4f (r) must vanish identically for the above equation to hold. One can to ~v∞ . Therefore, 4 ∇  

~ 4f (r) = ∇ ~ 4[4f (r)] , so that the checkfor instance using componentsthe identity 4 ∇ equation obeyed by f (r) becomes 4[4f (r)] = const. The integration constant must be zero, since it is a fourth derivative of f (r), while the velocity ~u(~r), which according to Eq. (VI23) depends on the second derivatives, must vanish as r ∞ One thus has 4[4f (r)] = 0. (34) ~ (r) ×~v∞ are both unsatisfactory: the velocity ~u(~r) is then always The simpler guesses ~u(~r) = f (r)~v∞ or ~u(~r) = ∇f parallel resp. orthogonal to ~v∞ , so that ~v(~r) cannot vanish everywhere at the surface of the sphere 99 VI.3 Flows at small Reynolds number In spherical coordinates, the Laplacian reads ∂2 `(` + 1) 2 ∂ 4= 2 + − , ∂r r ∂r r2 with ` an integer that depends on the angular dependence of the function: given the spherical symmetry of the problem for f ,

one should take ` = 0. Making the ansatz 4f (r) = C/rα , the equation 4[4f (r)] = 0 is only satisfied for α = 0 or 1. Using Eq (VI23) and the condition ~u(~r) ~0 for r ∞, only α = 1 is possible. The general solution of the linear differential equation 4f (r) = C d2 f (r) 2 df (r) = + 2 dr r dr r (VI.24a) is then given by B C + r, (VI.24b) r 2 where the first two terms in the right member represent the most general of the associated homogeneous equation, while the third term is a particular solution of the inhomogeneous equation. Equations (VI.23) and (VI24) lead to the velocity field

   ~v∞ − 3 ~er ·~v∞ ~er ~v∞ − ~er ·~v∞ ~er C C C C ~ r ~ r ~ −B + ~u(~r) = ∇ ·~v∞ − ~v∞ = −B + − ~v∞ 3 3 r 2r r r 2 r r

~v∞ − 3 ~er ·~v∞ ~er C ~v∞ + ~er ·~v∞ ~er = −B − . 3 r 2 r f (r) = A + The boundary condition ~u(|~r| = R) = −~v∞ at the surface of the sphere translates into    
C C B 3B ~v∞ + − 1− 3 − ~er ·~v∞

~er = ~0. 3 R 2R R 2R This must hold for any ~er , which requires B = R3 /4 and C = 6B/R2 = 3R/2, leading to
 R3 
 3R  ~v∞ + ~er ·~v∞ ~er − 3 ~v∞ − 3 ~er ·~v∞ ~er . ~v(~r) =~v∞ − 4r 4r (VI.25) Inserting this flow velocity in the Stokes equation (VI.17) gives the pressure 3 2 P (~r) = ηR ~er ·~v∞ + const. r2 With its help, one can then compute the mechanical stress (III.28) at a point on the surface of the sphere. The total force exerted by the flow on the latter follows from integrating the mechanical stress over the whole surface, and equals F~ = 6πRη~v∞ . (VI.26) This result is referred as Stokes’ law . Inverting the point of view, a sphere moving with velocity ~vsphere in a fluid at rest undergoes a friction force −6πRη~vsphere . Remarks: ∗ For the potential flow of a perfect fluid past a sphere with radius R, the flow velocity is(35)
 R3  ~v(~r) =~v∞ + 3 ~v∞ − 3 ~er ·~v∞ ~er . 2r That is, the velocity varies much faster

in the vicinity of the sphere than for the Stokes flow (VI.25): in the latter case, momentum is transported not only convectively but also by viscosity, which redistributes it over a wider region. The approximation of a flow at small Reynolds number, described by the Stokes equation, actually only holds in the vicinity of the sphere. Far from it, the flow is much less viscous (35) The proof can be found e.g in Landau–Lifshitz [3, 4] § 10 problem 2 100 Non-relativistic dissipative flows ∗ In the limit η 0, corresponding to a perfect fluid, the force (VI.26) exerted by the flow on the sphere vanishes: this is again the d’Alembert paradox encountered in § IV.43 c ∗ The proportionality factor between the sphere velocity and the friction force it experiences is called the mobility (lxiv) µ. According to Stokes’ law (VI26), for a sphere in the creeping-flow regime one has µ = 1/(6πRη). In his famous article on Brownian motion [26], A. Einstein related this mobility

with the diffusion coefficient D of a suspension of small spheres in a fluid at rest: D = µkB T = kB T . 6πRη This formula (Stokes–Einstein equation) was checked experimentally by J. Perrin, which allowed him to determine a value of the Avogadro constant and to prove the “discontinuous structure of matter” [27]. VI.4 Boundary layer The Reynolds number defined in Sec. VI21, which quantifies the relative importances of viscous and inertial effects in a given flow, involves characteristic length and velocity scales Lc , vc , that depend on the geometry of the fluid motion. When the flow involves an obstacle, as was the case in the example presented in Sec. VI32, a natural choice when studying the details of the fluid motion in the vicinity of the obstacle is to adopt the typical size R of the latter as characteristic length Lc defining the Reynolds number. Far from the obstacle, however, it is no longer obvious that R is really relevant. For Lc , a better choice might be the

distance to the obstacleor to any other wall or object present in the problem. Such a characteristic length gives a Reynolds number which can be orders of magnitude larger than the value computed with Lc . That is, even if the flow is viscous (small Re) close to the obstacle, far from it the motion could still be to a large extent inviscid (large Re) and thus well approximated by a perfect-fluid description. The above argumentation suggests that viscous effects may not be relevant throughout the whole fluid, but only in the region(s) in the vicinity of walls or obstacles. This is indeed the case, and the corresponding region surrounding walls or obstacles is referred to as boundary layer .(lxv) In the latterwhich often turns out to be rather thin, the velocity grows rapidly from its vanishing value at the surface of objects (no-slip condition) to the finite value it takes far from them and which is mostly imposed by the boundary conditions “at infinity”. In this Section, we shall

first illustrate on an example flow the existence of the boundary layer, computing in particular its typical width (Sec. VI41) The latter can also be estimated in a more general approach to the description of the fluid motion inside the boundary layer, as sketched in Sec. VI42 VI.41 Flow in the vicinity of a wall set impulsively in motion Consider an incompressible Newtonian fluid with uniform kinematic shear viscosity ν situated in the upper half-space y > 0, at rest for t < 0. The volume forces acting on the fluid are supposed to be negligible. At t = 0, the plane y = 0 is suddenly set in uniform motion parallel to itself, with constant velocity ~u(t > 0) = u~ex . As a consequence, the fluid in the vicinity of the plane is being dragged along; thanks to the viscous forces, the motion is transfered to the next fluid layers. The resulting flow is assumed to be laminar, with a fluid velocity parallel to ~ex . (lxiv) Beweglichkeit, Mobilität (lxv) Grenzschicht 101

VI.4 Boundary layer The invariance of the problem geometry under translations in the x- or z-directions justifies an ansatz ~v(t,~r) = v(t, y)ex which automatically fulfills the incompressibility condition, and similarly for the pressure field. That is, there are no gradients along the x- and z-directions As a result, the incompressible Navier–Stokes equation (III.32), projected onto the x-direction, reads ∂ v(t, y) ∂ 2 v(t, y) =ν . ∂t ∂y 2 (VI.27a) The boundary conditions are on the one hand the no-slip requirement at the moving plane, namely v(t, y = 0) = u for t > 0; (VI.27b) on the other hand, the fluid infinitely far from the moving plane remains unperturbed, i.e lim v(t, y) = 0 for t > 0. y∞ (VI.27c) Eventually, there is the initial condition v(t = 0, y) = 0 ∀y > 0. (VI.27d) The equations governing the motion (VI.27) involve only two dimensionful quantities, namely the plane velocity u and the fluid kinematic viscosity ν. With their help, one

can construct a characteristic time ν/u2 and a characteristic length ν/u in a unique manner, up to numerical factors. Invoking dimensional arguments, one thus sees that the fluid velocity is necessarily of the form   2 u t uy , , v(t, y) = uf 1 ν ν with f1 a dimensionless function of dimensionless variables. Since t and y are independent, so are their reduced versions u2 t/ν and uy/ν. Instead of the latter, one may adopt the equivalent set √ 2 u t/ν, ξ ≡ y/(2 νt), i.e write  2  y u t v(t, y) = uf 2 , √ , ν 2 νt with f2 again a dimensionless function. The whole problem (VI.27) is clearly linear in u, since the involved differential equations continuity equation and Navier–Stokes equation (VI.27a)are linear; this allows us to exclude any dependence of f2 on the variable u2 t/ν, so that the solution is actually of the form   y v(t, y) = uf √ (VI.28) 2 νt with f dimensionless, i.e dependent on a single reduced variable Inserting the latter ansatz in Eq. (VI27a)

leads after some straightforward algebra to the ordinary differential equation f 00 (ξ) + 2ξ f 0 (ξ) = 0, (VI.29a) with f 0 , f 00 the first two derivatives of f. Meanwhile, the boundary conditions (VI27b)–(VI27c) become f(0) = 1 , lim f(ξ) = 0. (VI.29b) ξ∞ The corresponding solution is f(ξ) = erfc(ξ) = 1 − erf(ξ), (VI.30) as(36) where erf denotes the (Gauss) error function, defined Z ξ 2 2 erf(ξ) ≡ √ e−υ dυ π 0 (36) (VI.31) The reader interested in its properties can have a look at the NIST Handbook of mathematical functions [28], Chapter 7. 102 Non-relativistic dissipative flows while erfc is the complementary error function (36) 2 erfc(ξ) ≡ √ π Z ∞ 2 e−υ dυ. (VI.32) ξ All in all, the solution of the problem (VI.27) is thus    y . v(t, y) = u 1 − erf √ 2 νt (VI.33) For ξ = 2, erf(2) = 0.99532 , ie erfc(2) ≈ 0005 That is, at given t, the magnitude of the velocity at √ y = δl (t) ≡ 4 νt (VI.34) is reduced

by a factor 200 with respect to its value at the moving plane. This length δl (t) is a typical measure for the width of the boundary layer over which momentum is transported from the plane into the fluid, i.e the region in which the fluid viscosity plays a role The width (VI.34) of the boundary layer increases with the square root of time: this is the typical behavior expected for a diffusive processwhich is understandable since Eq. (VI27a) is nothing but the classical diffusion equation. Remark: The above problem is often referred to as first Stokes problem or Rayleigh problem.(al) In the second Stokes problem, the plane is not set impulsively into motion, it oscillates sinusoidally in its own plane with a frequency ω. In that case, the amplitude of the induced fluid oscillations decrease “only” exponentially withp the distance to the plane, and the typical extent of the region affected by shear viscous effects is ν/ω. VI.42 Modeling of the flow inside the boundary layer As

argued in the introduction to the present Section, the existence of a “small” boundary layer, to which the effects induced by viscosity in the vicinity of an obstaclemore specifically, the influence of the no-slip condition at the boundariesare confined, can be argued to be a general feature. Taking its existence as grantedwhich is not necessary the case for every flow, we shall now model the fluid motion inside such a boundary layer. For simplicity, we consider a steady incompressible two-dimensional flow past an obstacle of typical size Lc , in the absence of relevant volume forces. At each point of the surface of the obstacle, the curvature radius is assumed to be large with respect to the local width δl of the boundary layer. That is, using local Cartesian (x, y) coordinates with x resp. y parallel resp orthogonal to the surface, the boundary layer has a large sizeof order Lc along the x-direction, while it is much thinnerof order δl along y. For the sake of brevity, the

variables (x, y) of the various dynamical fields vx , vy , P will be omitted. For the fluid inside the boundary layer, the equations of motion are on the one hand the incom~ ·~v = 0, i.e pressibility condition ∇ ∂ vx ∂ vy + = 0; (VI.35a) ∂x ∂y on the other hand, the incompressible Navier–Stokes equation (III.32), projected on the x- and y-axes, gives    2  ∂ ∂ 1 ∂P ∂2 ∂ vx + vy vx = − +ν + vx , (VI.35b) ∂x ∂y ρ ∂x ∂x2 ∂y 2  2    ∂ ∂ 1 ∂P ∂ ∂2 vx + vy vy = − +ν + vy . (VI.35c) ∂x ∂y ρ ∂y ∂x2 ∂y 2 (al) J. W Strutt, Lord Rayleigh, 1842–1919 103 VI.4 Boundary layer Since the boundary layer is much extended along the tangential direction than along the normal one, the range of x values is much larger than that of y values. To obtain dimensionless variables taking their values over a similar interval, one defines x y x∗ ≡ , y∗ ≡ (VI.36) Lc δl where the typical extent in the normal direction, i.e the width

of the boundary layer δl  Lc (VI.37) has to be determined be requiring that both x∗ , y ∗ are of order unity. Remarks: ∗ In realistic cases, the width δl may actually depend on the position x along the flow boundary, yet this complication is ignored here. ∗ If the local radius of curvature of the boundary is not much larger than the width δl of the boundary layer, one should use curvilinear coordinates x1 (tangential to the boundary) and x2 instead of Cartesian ones, yet within that alternative coordinate system the remainder of the derivation still holds. Similarly, the dynamical fields are rescaled to yield dimensionless fields: vx∗ ≡ vx v∞ , vy∗ ≡ vy u , P∗ ≡ P 2 ρv∞ , (VI.38) where, in order to account for the expectation that the normal velocity vy is (in average) much smaller than the tangential one vx which is of order v∞ at the outer edge of the boundary layer, a second velocity scale u  v∞ , (VI.39) was introduced, such that vx∗

, vy∗ , and P ∗ are of order unity. These fields are functions of the dimensionless variables (x∗ , y ∗ ), although this shall not be written explicitly. Eventually, the Reynolds number corresponding to the motion along x is Re ≡ Lc v∞ . ν (VI.40) With the help of definitions (VI.36)–(VI40), the equations of motion (VI35) can be recast in a dimensionless form: Lc u ∂ vy∗ ∂ vx∗ + = 0; (VI.41a) ∂x∗ δl v∞ ∂y ∗   ∗ δ2l ∂ 2 vx∗ Lc u ∗ ∂ vx∗ ∂P ∗ 1 L2c ∂ 2 vx∗ ∗ ∂ vx vx ∗ + v (VI.41b) =− ∗ + + 2 ∗2 , ∂x δl v∞ y ∂y ∗ ∂x Re δ2l ∂y ∗2 Lc ∂x   δ2l ∂ 2 vy∗ u ∗ ∂ vy∗ Lc u2 ∗ ∂ vy∗ Lc ∂ P ∗ 1 L2c u ∂ 2 vy∗ v v , (VI.41c) + =− + + 2 y ∂y ∗ v∞ x ∂x∗ δl v∞ δ ∂y ∗ Re δ2l v∞ ∂y ∗2 L2c ∂x∗2 Consider first the continuity equation (VI.41a) It will only yield a non-trivial constraint on the flow if both terms have the same order of magnitude, which is possible

if Lc u = 1, δl v ∞ (VI.42) yielding a first condition on the unknown characteristic quantities δl and u. In turn, a second constraint comes from the dimensionless Navier–Stokes equation (VI.41b) along the tangential direction. In the boundary layer, by definition, the effects from inertia encoded in the convective term and those of viscosity are of the same magnitude, which necessitates that 104 Non-relativistic dissipative flows the prefactor of the viscous term be of order unity. This suggests the condition L2c 1 = 1. δ2l Re (VI.43) Equations (VI.42)–(VI43) are then easily solved, yielding for the unknown quantities characterizing the flow along the direction normal to the boundary v∞ Lc , u= √ . (VI.44) δl = √ Re Re As in the first or second Stokes problems, see e.g Eq (VI34), the width of the boundary layer is proportional to the square root of the kinematic viscosity ν. Substituting the conditions (VI.42)–(VI43) in the system of equations (VI41) and

keeping only the leading terms, one eventually obtains ∂ vy∗ ∂ vx∗ + = 0; (VI.45a) ∂x∗ ∂y ∗ ∂ P ∗ ∂ 2 v∗ ∂ v∗ ∂ v∗ (VI.45b) vx∗ x∗ + vy∗ x∗ = − ∗ + ∗2x , ∂x ∂y ∂x ∂y ∂P ∗ = 0. (VI.45c) ∂y ∗ These equations constitute the simplified system, first by written down by Prandtl,(am) that describe the fluid motion in a laminar boundary layerwhere the laminarity assumption is hidden in the use of the typical length scale Lc imposed by geometry, rather than of a smaller one driven by turbulent patterns. VI.5 Vortex dynamics in Newtonian fluids The equations derived in Sec. IV32 regarding the behavior of vorticity in a perfect fluid are easily generalized to the case of a Newtonian fluid. VI.51 Vorticity transport in Newtonian fluids As was done with the Euler equation when going from Eq. (III18) to the
Eq (III20),
one may ~ ~v2 +~v × ω ~ ~v = 1 ∇ ~. rewrite the convective term in the Navier–Stokes equation (III.31) as ~v ·

∇ 2 ~ ~ Assuming then that the volume forces are conservative, i.e fV = −ρ∇Φ, and taking the rotational curl, one easily finds ~ P (t,~r) × ∇ρ(t,~ ~   ∂~ ω (t,~r) ~ ∇ r) − ∇ × ~v(t,~r) × ω + ν4~ ω (t,~r), (VI.46) ~ (t,~r) = − 2 ∂t ρ(t,~r) which generalizes Eq. (IV20) to the case of Newtonian fluids Note that even without assuming that the flow is incompressible, the term involving the bulk viscosity has already dropped out from the problem. As in Sec. IV32, the second term in the left member can be further transformed, which leads to the equivalent forms ~ P (t,~r) × ∇ρ(t,~ ~    D~ ω (t,~r)  ∇ r) ~ ~v(t,~r) − ∇ ~ ·~v(t,~r) ω = ω ~ (t,~r) · ∇ + ν4~ ω (t,~r), (VI.47a) ~ (t,~r) − 2 Dt ρ(t,~r) involving the material derivative D~ ω /Dt, or else ~ P (t,~r) × ∇ρ(t,~ ~   ∇ r) D~v ω ~ (t,~r) ~ ·~v(t,~r) ω ~ (t,~r) − =− ∇ + ν4~ ω (t,~r), (VI.47b) 2 Dt ρ(t,~r) which makes use of the comoving time-derivative (IV.22a)

(am) L. Prandtl, 1875–1953 105 VI.5 Vortex dynamics in Newtonian fluids The right hand side of this equation simplifies in various cases. In the particular of a barotropic fluid, the second term vanishes. In an incompressible flow, the first two terms are zero As we shall illustrate on an example, the viscous term, proportional to the Laplacian of vorticity, is of diffusive nature, and tends to spread out the vorticity lines over a larger region. VI.52 Diffusion of a rectilinear vortex As example of application of the equation of motion introduced in the previous Section, let us consider the two-dimensional motion in the (x, y)-plane of an incompressible Newtonian fluid with conservative forces, in which there is at t = 0 a rectilinear vortex along the z-axis: Γ0 ω ~ (t = 0,~r) = δ(z)~ez (VI.48) 2πr with r the distance from the z-axis. Obviously, the circulation around any curve circling this vortex once is simply Γ0 . At time t > 0, this vortex will start diffusing,

with its evolution governed by Eq. (VI46) Given the symmetry of the problem round the z-axis, which suggests the use of cylindrical coordinates (r, θ, z), the vorticity vector will remain parallel to ~ez , and its magnitude should only depend on r: ω ~ (t,~r) = ω z (t, r)~ez (VI.49) This results in a velocity field ~v(t,~r) in the (x, y)-plane, in the orthoradial direction. As a consequence the convective derivative in the left hand side of Eq. (VI47a) vanishes since ω ~ (t, r) has no
~ ~v also vanishes, since the velocity is independent of z. gradient along eθ . Similarly, the term ω ~ ·∇ ~ ·~v vanishes thanks to the assumed incompressibility. All in all, Eventually, the term involving ∇ the vorticity thus obeys the diffusion equation  2 z  ∂ω z (t, r) ∂ ω (t, r) 1 ∂ω z (t, r) z = ν4ω (t, r) = ν + , (VI.50) ∂t ∂r2 r ∂r with the initial condition (VI.48) The problem is clearly linear in Γ0 , so that the solution ω z (t, r) should be proportional to

Γ0 , without any further dependence on Γ0 . This leaves the kinematic viscosity ν as only dimensionful parameter available in the problem: using a dimensional reasoning similar to that made in the study of the first Stokes problem (Sec. VI41), there is a single relevant dimensionless variable, namely ξ = r2 /(νt), combining the time and space variables. The only ansatz respecting the dimensional requirements is then Γ0 r2 ω z (t, r) = f(ξ), with ξ ≡ (VI.51) νt νt with f a dimensionless function. Inserting this ansatz into Eq (VI50) leads to the ordinary differential equation  f(ξ) + ξ f 0 (ξ) + 4 f 0 (ξ) + ξ f 00 (ξ)] = 0. (VI.52) A first integration yields ξ f(ξ) + 4ξ f 0 (ξ) = const. In order to satisfy the initial condition, the integration constant should be zero, leaving with the linear differential equation f(ξ) + 4f 0 (ξ) = 0, which is readily integrated to yield f(ξ) = C e−ξ/4 , that is Γ0 2 C e−r /(4νt) , νt with C an integration which still

has to be fixed. ω z (t, r) = (VI.53) 106 Non-relativistic dissipative flows To determine the latter, let us consider the circulation of the velocity at time t around a circle CR of radius R centered on the axis z = 0: Z R Z 2π Z R I ~v(t,~r) · d~` = ω z (t, r) r dr dθ = 2π ω z (t, r) r dr dθ, (VI.54) Γ(t, R) = CR 0 0 0 where the second identity follows from the Stokes theorem while the third is trivial. Inserting the solution (VI.53) yields   2 Γ(t, R) = 4πΓ0 C 1 − e−R /(4νt) , showing the C should equal 1/4π to yield the proper circulation at t = 0. All in all, the vorticity field in the problem reads Γ0 −r2 /(4νt) ω z (t, r) = e . (VI.55) 4πνt √ That is, the vorticity extends over a region of typical width δ(t) = 4νt, √ which increases with time: √ one recognizes the characteristic diffusive behavior, proportional to tas well as the typical ν dependence of the size of the region affected by viscous effects, encountered in Sec. VI4 The

vorticity (VI.55) leads to the circulation around a circle of radius R   2 (VI.56) Γ(t, R) = Γ0 1 − e−R /(4νt) , which at given R decreases with time, in contrast to the perfect-fluid case, in which the circulation would be conserved. Eventually, one can also easily compute the velocity field associated with the expanding vortex, namely ~eθ Γ0  2 ~v(t,~r) = 1 − e−r /(4νt) , (VI.57) 2πr r where |~eθ | = r. VI.6 Absorption of sound waves Until now, we only investigated incompressible motions of Newtonian fluids, in which the bulk viscosity can from the start play no role. The simplest example of compressible flow is that of sound waves, which were already studied in the perfect-fluid case. As in Sec V11, we consider small adiabatic perturbations of a fluid initially at rest and with uniform propertieswhich implies that external volume forces like gravity are neglected. Accordingly, the dynamical fields characterizing the fluid are ρ(t,~r) = ρ0 + δρ(t,~r), P

(t,~r) = P 0 + δ P (t,~r), ~v(t,~r) = ~0 + δ~v(t,~r), (VI.58a) with |δρ(t,~r)|  ρ0 , |δ P (t,~r)|  P 0 , δ~v(t,~r)  cs , (VI.58b) where cs denotes the quantity which in the perfect-fluid case was found to coincide with the phase velocity of similar small perturbations, i.e the “speed of sound”, defined by Eq (V5)   ∂P 2 . (VI.58c) cs ≡ ∂ρ S,N As in Sec. V11, this relation will allow us to relate the pressure perturbation δ P to the variation of mass density δρ. Remark: Anticipating on later findings, the perturbations must actually fulfill a further condition, related to the size of their spatial variations [cf. Eq (VI68)] This is nothing but the assumption of “small gradients” that underlies the description of their propagation with the Navier–Stokes equation, i.e with first-order dissipative fluid dynamics 107 VI.6 Absorption of sound waves For the sake of simplicity, we consider a one-dimensional problem, i.e perturbations propagating

along the x-direction and independent of y and zas are the properties of the underlying background fluid. Under this assumption, the continuity equation (III9) reads ∂δv(t, x) ∂ρ(t, x) ∂ρ(t, x) + ρ(t, x) + δv(t, x) = 0, ∂t ∂x ∂x while the Navier–Stokes equation (III.31) becomes     2 ∂δ P (t, x) ∂δv(t, x) ∂δv(t, x) 4 ∂ δv(t, x) ρ(t, x) =− + δv(t, x) + η+ζ . ∂t ∂x ∂x 3 ∂x2 (VI.59a) (VI.59b) Substituting the fields (VI.58a) in these equations and linearizing the resulting equations so as to keep only the leading order in the small perturbations, one finds ∂δρ(t, x) ∂δv(t, x) + ρ0 = 0, ∂t ∂x   2 ∂δ P (t, x) ∂δv(t, x) 4 ∂ δv(t, x) ρ0 =− + η+ζ . ∂t ∂x 3 ∂x2 (VI.60a) (VI.60b) In the second equation, the derivative ∂(δ P )/∂x can be replaced by c2s ∂(δρ)/∂x. Let us in addition introduce the (traditional) notation(37)   1 4 η+ζ , (VI.61) ν̄ ≡ ρ0 3 so that Eq. (VI60b) can be rewritten as

ρ0 ∂δρ(t, x) ∂ 2 δv(t, x) ∂δv(t, x) + c2s = ρ0 ν̄ . ∂t ∂x ∂x2 (VI.62) Subtracting c2s times the time derivative of Eq. (VI60a) from the derivative of Eq (VI62) with respect to x and dividing the result by ρ then yields 2 ∂ 2 δv(t, x) ∂ 3 δv(t, x) 2 ∂ δv(t, x) − c = ν̄ . s ∂t2 ∂x2 ∂t ∂x2 (VI.63a) One easily checks that the mass density variation obeys a similar equation 2 ∂ 3 δρ(t, x) ∂ 2 δρ(t, x) 2 ∂ δρ(t, x) − c = ν̄ . s ∂t2 ∂x2 ∂t ∂x2 (VI.63b) In the perfect-fluid case ν̄ = 0, one recovers the traditional wave equation (V.9a) Equations (VI.63) are homogeneous linear partial differential equations, whose solutions can be written as superposition of plane waves. Accordingly, let us substitute the Fourier ansatz e ~k) e−iωt+i~k·~r δρ(t,~r) = δρ(ω, (VI.64) in Eq. (VI63b) This yields after some straightforward algebra the dispersion relation ω 2 = c2s k 2 − iωk 2 ν̄. (VI.65) Obviously, ω and k

cannot be simultaneously real numbers. Let us assume k ∈ R and ω = ωr + iωi , where ωr , ωi are real. The dispersion relation becomes ωr2 − ωi2 + 2iωr ωi = c2s k 2 − iωr k 2 ν̄ + ωi k 2 ν̄, (37) Introducing the kinetic shear resp. bulk viscosity coefficients ν resp ν 0 , one has ν̄ = 34 ν + ν 0 , hence the notation 108 Non-relativistic dissipative flows which can only hold if both the real and imaginary parts are equal. The identity between the imaginary parts reads (for ωr 6= 0) 1 ωi = − ν̄k 2 , (VI.66) 2 which is always negative, since ν̄ is non-negative. This term yields in the Fourier ansatz (VI64) 2 an exponentially decreasing factor e−i(iωi )t = e−ν̄k t/2 which represents the damping or absorption of the sound wave. The perturbations with larger wave number k, ie corresponding to smaller length scales, are more dampened that those with smaller k. This is quite natural, since a larger k also means a larger gradient, thus an

increased influence of the viscous term in the Navier–Stokes equation. In turn, the identity between the real parts of the dispersion relation yields 1 ωr2 = c2s k 2 − ν̄ 2 k 4 . 4 (VI.67) This gives for the phase velocity cϕ ≡ ω/k of the traveling waves 1 c2ϕ = c2s − ν̄ 2 k 2 . 4 (VI.68) That is, the “speed of sound” actually depends on its wave number k, and is smaller for small wavelength, i.e high-k, perturbationswhich are also those which are more damped Relation (VI.68) also shows that the whole linear description adopted below Eqs (VI59) requires that the perturbations have a relatively large wavelength, namely k  2cs /ν̄, so that cϕ remain realvalued. This is equivalent to stating that the dissipative term ν̄4δv ∼ k 2 ν̄δv in the Navier–Stokes equation should be much smaller than the local acceleration ∂t δv ∼ ωδv ∼ cs kδv. Remarks: ∗ Instead of considering “temporal damping” as was done above by assuming k ∈ R but ω ∈

C, one may investigate “spatial damping”, i.e assume ω ∈ R and put the whole complex dependence in the wave number k = kr + iki . For (angular) frequencies ω much smaller than the inverse of the typical time scale τν ≡ ν̄/c2s , one finds     3 2 2 ω 3 2 2 2 2 2 ⇔ cϕ ≡ cs 1 + ω τν ω cs kr 1 + ω τν 4 kr 8 i.e the phase velocity increases with the frequency, and on the other hand ki ν̄ω 2 . 2c3s (VI.69) The latter relation is known as Stokes’ law of sound attenuation, ki representing the inverse of the typical distance over which the sound wave amplitude decreases, due to the factor ei(iki )x = e−ki x in the Fourier ansatz (VI.64) Larger frequencies are thus absorbed on a smaller distance from the source of the sound wave. Substituting k = kr + iki = kr (1 + iκ) in the dispersion relation (VI.65) and writing the identity of the real and imaginary parts, one obtains the system ( 2κ = ωτν (1 − κ 2 ) ω 2 = c2s kr2 (1 + 2ωτν κ − κ 2

) The first equation is a quadratic equation in κ that admits one positive
and one negative solution: the latter can be rejected, while the former is κ ωτν /2 + O (ωτν )2 . Inserting it in the second equation leads to the wanted results.  An exact solution of the system of equations exists, yes it is neither enlightening mathematically, nor relevant from the physical point of view in the general case, as discussed in the next remark. VI.6 Absorption of sound waves 109 One may naturally also analyze the general case in which both ω and k are complex numbers. In any case, the phase velocity is given by cϕ ≡ ω/kr , although it is more difficult to recognize the physical content of the mathematical relations. ∗ For air or water, the reduced kinetic viscosity ν̄ is of order 10−6 –10−5 m2 · s−1 . With speeds of sound cs 300–1500 m · s−1 , this yields typical time scales τν of order 10−12 –10−10 s. That is, the change in the speed of sound

(VI.68), or equivalently deviations from the assumption ωτν  1 underlying the attenuation coefficient (VI69), become relevant for sound waves in the gigahertz/terahertz regime(!). This explains why measuring the bulk viscosity is a non-trivial task The wavelengths cs τν corresponding to the above frequencies τν−1 are of order 10−9 –10−7 m. This is actually not far from the value of the mean free path in classical fluids, so that the whole description as a continuous medium starts being questionable. Bibliography for Chapter VI • National Committee for Fluid Mechanics film & film notes on Rotating flows, Low Reynolds Number Flow , Fundamentals of Boundary Layers and Vorticity; • Faber [1] Chapters 6.6, 69 and 611; • Guyon et al. [2] Chapters 45, 732, 9 & 101–104; • Landau–Lifshitz [3, 4] Chapter II § 17–20 & 24, Chapter IV § 39 and Chapter VIII § 79; • Sommerfeld [5, 6] Chapters II § 10, III § 16 and VII § 35. C HAPTER VII

Turbulence in non-relativistic fluids All examples of flows considered until now in these notes, either of perfect fluids (Chapters IV and V) or of Newtonian ones (Chapter VI), share a common property, namely they are all laminar. This assumptionwhich often translates into a relative simplicity of the flow velocity profileis however not the generic case in real flows, which most often are to some more or less large extent turbulent. The purpose of this Chapter is to provide an introduction to the problematic of turbulence in fluid motions. A number of experiments, in particular those conducted by O. Reynolds, have hinted at the possibility that turbulence occurs when the Reynolds number (VI.12) is large enough in the flow, i.e when convective effects predominate over the shear viscous ones that drive the mean fluid motion over which the instabilities develop. This distinction between mean flow and turbulent fluctuations can be modeled directly by splitting the dynamical fields into two

parts, and one recovers with the help of dimensional arguments the role of the Reynolds number in separating two regimes, one in which the mean viscous flow dominates and one in which turbulence takes over (Sec. VII1) Despite its appeal, the decomposition into a mean flow and a turbulent motion has the drawback that it leads to a system of equations of motion which is not closed. A possibility to remedy this issue is to invoke the notion of a turbulent viscosity, for which various models have been proposed (Sec. VII2) Even when the system of equations of motion is closed, it still involves averageswith an a priori unknown underlying probability distribution. That is, the description of turbulent part of the motion necessitates the introduction of a few concepts characterizing the statistics of the velocity field (Sec. VII3) For the sake of simplicity, we shall mostly consider turbulence in the three-dimensional incompressible motion of Newtonian fluids with constant and homogeneous

properties (mass density, viscosity. ), in the absence of relevant external bulk forces, and neglecting possible temperature gradientsand thereby convective heat transport. VII.1 Generalities on turbulence in fluids In this Section, a few experimental facts on turbulence in fluid flows is presented, and the first steps towards a modeling of the phenomenon are introduced. VII.11 Phenomenology of turbulence VII.11 a Historical example: Hagen–Poiseuille flow The idealized Hagen–Poiseuille flow of a Newtonian fluid in a cylindrical tube was already partly discussed in Sec. VI14 It was found that in the stationary laminar regime in which the velocity field ~v is purely parallel to the walls of the tube, the mass flow rate Q across the cylinder cross section is given by the Hagen–Poiseuille law πρa4 ∆P Q=− , (VI.9) 8η L with a the tube radius, ∆P /L the pressure drop per unit length, and ρ, η the fluid properties. ::::::::::::::::::::::::::::::::::::::::::::::::::::

111 VII.1 Generalities on turbulence in fluids Due to the viscous friction forces, part of the kinetic energy of the fluid motion is transformed into heat. To compensate for these “losses” and keep the flow in the stationary regime, energy has to be provided to the fluid, namely in the form of the mechanical work of the pressure forces driving the flow. Thus, the rate of energy dissipation per unit mass is(38) Ė diss. = − 1 ∆P 8νhvi2 hvi = ρ L a2 (VII.1) with hvi the average flow velocity across the tube cross section, hvi = Q a2 ∆P = − . πa2 ρ 8η L Thus, in the laminar regime the rate Ė diss. is proportional to the kinematic viscosity ν and to the square of the average velocity. According to the Hagen–Poiseuille law (VI.9), at fixed pressure gradient the average velocity hvi grows quadratically with the tube radius. In practice, the rise is actually slower, reflecting a higher rate of energy loss in the flow as given by the laminar prediction (VII.1)

Thus, the mean rate of energy dissipation is no longer proportional to hvi2 , but rather to a higher power of hvi. Besides, the flow velocity profile across the tube cross section is no longer parabolic, but (in average) flatter around the cylinder axis, with a faster decrease at the tube walls. VII.11 b Transition to a turbulent regime Consider a given geometrysay for instance that of the Hagen–Poiseuille flow or the motion of a fluid in a tube with fixed rectangular cross section. In the low-velocity regime, the flow in that geometry is laminar, and the corresponding state(39) is stable against small perturbations, which are damped by viscosity (see Sec. VI6) However, when the average flow velocity exceeds some critical value, while all other characteristics of the flow, in particular the fluid properties, are fixed, the motion cannot remain laminar. Small perturbations are no longer damped, but can grow by extracting kinetic energy from the “main”, regular part of the fluid

motion. As a consequence, instead of simple pathlines, the fluid particles now follow more twisted ones: the flow becomes turbulent. In that case, the velocity gradients involved in the fluid motion are in average much larger than in a laminar flow. The amount of viscous friction is thus increased, and a larger fraction of the kinetic energy is dissipated as heat. ::::::::::::::::::::::::::::::::::::::::: The role of a critical parameter in the onset of turbulence was discovered by Reynolds in the case of the Hagen–Poiseuille flow of water, in which he injected some colored water on the axis of the tube, repeating the experiment for increasing flow velocities [19]. In the laminar regime found at small velocities, the streakline formed by the colored water forms a thin band along the tube axis, which does not mix with the surrounding water. Above some flow velocity, the streakline remains straight along some distance in the tube, then suddenly becomes instable and fills the whole

cross section of the tube. As Reynolds understood himself by performing his experiments with tubes of various diameters, the important parameter is not the velocity itself, but rather the Reynolds number Re (VI.12), which is proportional to the velocity. Thus, the transition to turbulence in flows with shear occurs at a “critical value” Rec , which however depends on the geometry of the flow. For instance Rec is of order 2000 for the Hagen–Poiseuille flow, but becomes of order 1000 for the plane Poiseuille flow investigated in Sec. VI13, while Rec 370 for the plane Couette flow (Sec VI12) (38) In this Chapter, we shall only discuss incompressible flows at constant mass density ρ, and thus always consider energies per unit mass. (39) This term really refers to a macroscopic “state” of the system in the statistical-physical sense. In contrast to the global equilibrium states usually considered in thermostatics, it is here a non-equilibrium steady state, in which local

equilibrium holds at every point. 112 Turbulence in non-relativistic fluids The notion of a critical Reynolds number separating the laminar and turbulent regimes is actually a simplification. In theoretical studies of the stability of the laminar regime against linear perturbations, such a critical value Rec can be computed for some very simple geometries, yielding e.g Rec = 5772 for the plane Poiseuille flow Yet the stability sometimes also depends on the size of the perturbation: the larger it is, the smaller the associated critical Rec is, which hints at the role of nonlinear instabilities. In the following, we shall leave aside the problem of the temporal onset of turbulenceand thereby of the (in)stability of laminar flows, and focus on flows which are already turbulent when we start looking at them. VII.12 Reynolds decomposition of the fluid dynamical fields Since experiment as well as reasoning hint at the existence of an underlying “simple”, laminar flow over which

turbulence develops, a reasonable ansatz for the description of the turbulent motion of a fluid is to split the relevant dynamical fields into two components: a first one which varies slowly both in time t and position ~r, and a rapidly fluctuating component, which will be denoted with primed quantities. In the case of the flow velocity field~v(t,~r), this Reynolds decomposition (lxvi) reads [29] ~v(t,~r) = ~v(t,~r) +~v0(t,~r), (VII.2) with ~v resp. ~v0 the “slow” resp “fast” component For the pressure, one similarly writes P (t,~r) = P (t,~r) + P 0(t,~r). (VII.3) The fluid motion with velocity ~v and pressure P is then referred to as “mean flow”, that with the rapidly varying quantities as “fluctuating motion”. As hinted at by the notation, ~v(t,~r) represents an average, with some underlying probability distribution. Theoretically, the Reynolds average · should be an ensemble average, obtained from an infinitely large number of realizations, namely experiments

or computer simulations; in practice, however, there is only a finite number N of realizations ~v(n)(t,~r). If the turbulent flow is statistical stationary, one may invoke the ergodic assumption and replace the ensemble average by a time average: Z N 1 X (n) 1 t+T /2 0 ~v(~r) ≡ lim ~v (t,~r) ≈ ~v(t ,~r) dt0 , N ∞ N T t−T /2 n=1 with T much larger than the autocorrelation time of the turbulent velocity ~v0(t,~r). If the flow is not statistically stationary, so that ~v(t,~r) also depends on time, then T must also be much smaller than the typical time scale of the variations of the mean flow. Using the same averaging procedure, the fluctuating velocity must obey the condition ~v0(t,~r) = ~0. (VII.4) Despite this fact, the turbulent velocity ~v0(t,~r) still plays a role in the dynamics, in particular that of the mean flow, because its two-point, three-point and higher (auto)correlation functions are still in general non-vanishing. For instance, one can writeassuming that the

mass density ρ is constant and uniform ρ vi (t,~r) vj (t,~r) = ρ vi (t,~r) vj (t,~r) + ρ v0i (t,~r) v0j (t,~r). The first term of the right member corresponds to convective part of the momentum-flux density (lxvi) Reynolds-Zerlegung 113 VII.1 Generalities on turbulence in fluids of the mean flow, while the second one TRij(t,~r) ≡ ρ v0i (t,~r) v0j (t,~r), (VII.5) which is simply the ij-component of the rank 2 tensor TR (t,~r) ≡ ρ~v0(t,~r) ⊗~v0(t,~r), (VII.6) is due to the rapidly fluctuating motion. TR is called turbulent stress or Reynolds stress(lxvii) VII.13 Dynamics of the mean flow For the sake of simplicity, the fluid motion will from now on be assumed to be incompressible. ~ ·~v(t,~r) = 0, leads to Thanks to the linearity of the averaging process, the kinematic condition ∇ the two relations ~ ·~v0(t,~r) = 0. ~ · ~v(t,~r) = 0 and ∇ (VII.7) ∇ That is, both the mean flow and the turbulent motion are themselves incompressible. The total flow

velocity~v obeys the usual incompressible Navier–Stokes equation [cf. Eq (III32)]    ∂~v(t,~r)  ~ P (t,~r) + η4~v(t,~r), ~ ρ (VII.8) + ~v(t,~r) · ∇ ~v(t,~r) = −∇ ∂t from which the equations governing the mean and turbulent flows can be derived. For the sake of brevity, the variables (t,~r) of the various fields will be omitted in the following. VII.13 a Equations for the mean flow ::::::::::::::::::::::::::::::::::::: Inserting the Reynolds decompositions (VII.2)–(VII3) in the Navier–Stokes equation (VII8) and averaging with the Reynolds average · leads to the so-called Reynolds equation  

∂~v ~ ~v = −∇ ~ P + η4~v − ρ ~v0 · ∇ ~ ~v0. ρ + ~v · ∇ ∂t (VII.9a) To avoid confusion, this equation is also sometimes referred to as Reynolds-averaged Navier–Stokes equation. In terms of components in a given system of coordinates, this becomes, after dividing by the mass density ρ 3 X
1 dP ∂ vi dv0i v0j i ~ + ~v · ∇ v = − − + ν4vi

. (VII.9b) ∂t ρ dxi dxj j=1 These two equations involve the material derivative “following the mean flow” D ∂ ~ + ~v · ∇. ≡ ∂t Dt (VII.10) Using the incompressibility of the fluctuating motion, the rightmost term in Eq. (VII9a) can be rewritten as

~ ~v0 = −ρ∇ ~ · TR . ~ · ~v0 ⊗~v0 = −∇ −ρ ~v0 · ∇ The Reynolds equation can thus be recast in the equivalent form [cf. Eq (III24b)]
∂ ~ · TR , ~ · T = −∇ ρ~v + ∇ (VII.11) ∂t with T the momentum-flux density of the mean flow, given by [cf. Eqs (III26b), (III26e)] S T ≡ P g−1 + ρ~v ⊗ ~v − 2ηS i.e, component-wise (lxvii) Reynolds-Spannung (VII.12a) 114 Turbulence in non-relativistic fluids Sij , T ij ≡ P g ij + ρvi vj − 2ηS (VII.12b) with S the rate-of-shear tensor [Eq. (II15b)] for the mean flow, with components [cf Eq (II15d)]   dvj 2 ij ~ 1 dvi ij + − g ∇ · ~v , (VII.12c) S ≡ 2 dxj dxi 3 where the third term within the brackets actually vanishes due to

the incompressibility of the mean flow, Eq. (VII7) The form (VII.11) of the Reynolds equation emphasizes perfectly the role of the Reynolds stress, i.e the turbulent component of the flow, as “external” force driving the mean flow In particular, the off-diagonal terms of the Reynolds stress describe shear stresses, which will lead to the appearance of eddies in the flow. Starting from the
Reynolds equation, one can derive the equation governing the evolution of the 2 kinetic energy 21 ρ ~v associated with the mean flow, namely   

D ρ~v2 ~ · P ~v + TR − 2ηS S · ~v + TR − 2ηS S : S. = −∇ (VII.13) Dt 2
2 This equation is conventionally rather written in terms of the kinetic energy per unit mass k ≡ 21 ~v , in which case it reads 

Dk 1 ~ = −∇ · P ~v + ~v0 ⊗~v0 − 2ν S · ~v + ~v0 ⊗~v0 − 2ν S : S , (VII.14a) ρ Dt or component-wise  3  3 3   X  X Dk d 1 j X 0i 0j ij 0i v0j − 2ν S ij S . + =− P v + v v − 2ν S v v (VII.14b) i

ij dxj ρ Dt j=1 i=1 i,j=1 In either form, the physical meaning of each term is rather transparent: first comes the convective transport of energy in the mean flow, given by the divergence of the energy flux density, inclusive a term from the turbulent motion. The second term represents the energy which is “lost” to the mean flow, namely either because it is dissipated by the viscous friction forces (term in ν S : S ), or because it is transferred to the turbulent part of the motion (term involving the Reynolds stress). To prove Eq. (VII13), one should first average the inner product with ~v of the Reynolds equa
~ P and ~v· ~v0 · ∇ ~ ~v0 under consideration of the incompressibility tion (VII.9), and then rewrite ~v· ∇ condition (VII.7) Remark: While equations (VII.9) or (VII14) do describe the dynamics of the mean flow, they rely on the Reynolds stress, which is not yet specified by the equations. VII.13 b Description of the transition to the turbulent regime

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Turbulence takes place when the effects of Reynolds stress TR which represents a turbulent transport of momentumpredominates over those of the viscous stress tensor 2ρν S associated with the mean flow, i.e when the latter can no longer dampen the fluctuations corresponding to the former. Let vc resp. Lc denote a characteristic velocity resp length scale of the fluid motion Assuming that averageshere, a simple over the volume is meantover the flow yield the typical orders of magnitude * 3 + * 3 + 3 X X v νv2 2 c v0i v0j S ij ∼ and ∼ 2c , (VII.15) ν S ij S ij Lc Lc i,j=1 i,j=1 then in the turbulent regime the first of these terms is significantly larger than the second, which corresponds to having a large value of the Reynolds number Re ≡ vc Lc /ν [Eq. (VI12)] 115 VII.1 Generalities on turbulence in fluids In that situation, the equation (VII.14) describing the evolution of the kinetic energy of the mean

flow becomes 

Dk ~ · 1 P ~v + ~v0 ⊗~v0 · ~v + ~v0 ⊗~v0 : S , = −∇ (VII.16a) ρ Dt or component-wise  X  3 3 3 X Dk d 1 j X 0i 0j
Pv + v v vi + v0i v0j Sij . (VII.16b) =− dxj ρ Dt i,j=1 i=1 j=1 That is, the viscosity is no longer a relevant parameter for the dynamics of the mean flow. As already argued above, the first term in Eq. (VII16) represents the convective transport of energy in the mean flow, while the second, “mixed” term models the transfer of energy from the mean flow into the turbulent motion, and thus corresponds to the energy “dissipated” by the mean flow. Invoking the first relation in Eq (VII15), the rate of energy dissipation in the mean flow is + * 3 X v3 (VII.17) v0i v0j S ij ∼ c . Ė diss. = Lc i,j=1 This grows like the third power of the typical velocity, i.e faster than vc2 , as argued at the end of § VII.11 a for the turbulent regime of the Hagen–Poiseuille flow In addition, this energy dissipation rate is actually

independent of the properties (mass density, viscosity. ) of the flowing fluid: turbulence is a characteristic of the motion, not of the fluid itself. Eventually, the middle term in Eq. (VII17) must be negative, so that the energy really flows from the mean flow to the turbulent motion, not in the other direction! Remark: Looking naively at the definition of the Reynolds number, the limit of an infinitely large Re corresponds to the case of a vanishing shear viscosity, that is, to the model of a perfect fluid. As was just discussed, this is clearly not the case: with growing Reynolds number, i.e increasing influence of the turbulent motion, the number of eddies in the flow also increases, in which energy is dissipated into heat. In contrast, the kinetic energy is conserved in the flow of a perfect fluid The solution to this apparent paradox is simply that with increasing Reynolds number, the velocity gradients in the flow also increase. In the incompressible Navier–Stokes

equation, the growth of 4~v compensates the decrease of the viscosity ν, so that the corresponding term does not disappear and the Navier–Stokes equation does not simplify to the Euler equation. VII.14 Necessity of a statistical approach As noted above, the evolution equation for the mean flow involves the Reynolds stress, for which no similar equation has yet be determined. A first, natural solution is simply to write down the evolution equation for the turbulent velocity ~v0(t,~r), see Eq. (VII25) below Invoking then the identity  ∂~v0(t,~r) ∂  0 ∂~v0(t,~r) ⊗~v0(t,~r) + ρ~v0(t,~r) ⊗ , ρ~v (t,~r) ⊗~v0(t,~r) = ρ ∂t ∂t ∂t one can derive a dynamical equation for TR , the so-called Reynolds-stress equation (lxviii) , which in component form reads   ij   j i TRij T DT dT dv d dv dv0i dv0j jk ik R 0 0ij 0 0i jk 0 0j ik TR = −2 P S + k P v g + P v g +ρv0i v0j v0k −ν − TR +T −2η k . k k dxk dx dx dx dx dxk Dt (VII.18) (lxviii)

Reynolds-Spannungsgleichung 116 Turbulence in non-relativistic fluids Irrespective of the physical interpretation of each of the terms in this equation, an important issue is that the evolution of ρv0i v0j involves a contribution from the components ρv0i v0j v0k of a tensor of degree 3. In turn, the evolution of ρv0i v0j v0k involves the tensor with components ρv0i v0j v0k v0l , and so on: at each step, the appearance of a tensor of higher degree simply reflects the nonlinearity of the Navier–Stokes equation. All in all, the incompressible Navier–Stokes equation (VII.8) is thus equivalent to an infinite hierarchy of equations relating the successive n-point autocorrelation functions of the fluctuations of the velocity field. Any subset of this hierarchy is not closed and involves more unknown fields than equations. A closure prescription, based on some physical assumption, is therefore necessary, to obtain a description with a finite number of equations governing the

(lower-order) autocorrelation functions. Such an approach is presented in Sec VII2 An alternative possibility is to assume directly some ansatz for the statistical behavior of the turbulent velocity, especially for its general two-point autocorrelation function, of which the equaltime and position correlator v0i (t,~r)v0j (t,~r) is only a special case. This avenue will be pursued in Sec. VII3 VII.2 Model of the turbulent viscosity A first possibility to close the system of equations describing turbulence consists in using the phenomenological concept of turbulent viscosity, which is introduced in Sec. VII21, and for which various models are quickly presented in Sec. VII22–VII24 VII.21 Turbulent viscosity The basic idea underlying the model is to consider that at the level of the mean flow, effect of the “turbulent friction” is to redistribute momentum from the high mean-velocity regions to the ones in slower motion, in the form of a diffusive transport. Accordingly, the

traceless part of the turbulent Reynolds stress is dealt with like the corresponding part of the viscous stress tensor (III.26e), and assumed to be proportional to the rate-of-shear tensor of the mean flow (Boussinesq hypothesis (an) ):   S(t,~r), (VII.19a) T R (t,~r) − Tr T R (t,~r) g−1 (t,~r) ≡ −2ρνturb. (t,~r)S where the proportionality factor involves the (kinematic) turbulent viscosity or eddy viscosity (40) νturb , which a priori depends on time and position. In terms of components in a coordinate system, and replacing the Reynolds stress and its trace by their expressions in terms of the fluctuating velocity, this reads 1 ρ v0i (t,~r) v0j (t,~r) − ρ [~v0(t,~r)]2 g ij (t,~r) ≡ 2ρνturb (t,~r) S ij (t,~r). 3 (VII.19b) Using the ansatz (VII.19) and invoking the incompressibility of the mean flow, from which follows ~ · S = 1 4~v, the Reynolds equation (VII.9) can be rewritten as ∇ 2    ∂~v(t,~r)  P (t,~r) [~v0(t,~r)]2 ~ ~ + 2νeff (t,~r)4~v(t,~r), +

~v(t,~r) · ∇ ~v(t,~r) = −∇ + (VII.20) ∂t ρ 3 with the effective viscosity νeff (t,~r) = ν + νturb (t,~r), while the term in curly brackets may be seen as an effective pressure. (40) turbulente Viskosität, Wirbelviskosität (an) J. Boussinesq, 1842–1929 (VII.21) 117 VII.2 Model of the turbulent viscosity Even if the intrinsic fluid properties, in particular its kinematic viscosity ν, are assumed to be constant and uniform, this does not hold for the turbulent and effective viscosities νturb , νeff , because they model not the fluid, but also its flowwhich is time and position dependent. Either starting from Eq. (VII20) multiplied by ~v, or substituting the Reynolds stress with the ansatz (VII.19) in Eq (VII14), one can derive the equation governing the evolution of the kinetic energy of the mean flow. In particular, one finds that the dissipative term is 3 X Ė diss = 2νeff S : S = 2νeff S ij S ij . i,j=1 Comparing with the rightmost term in Eq. (VII14)

gives for the effective viscosity X X 2ν − v0i v0j S ij S ij S ij νeff = i,j 2 X S ij S ij i,j  2 i,j X S ij S ij = ν, i,j where the inequality holds in the turbulent regime. There thus follows νeff ≈ νturb  ν It has been argued that in plasmas, the turbulent viscosity νturb could in some regimes be negativeand of the same magnitude as ν, leading to an “anomalaous” effective viscosity νeff much smaller than ν [30, 31]. Remark: To emphasize the distinction with the turbulent viscosity, ν is sometimes referred to as “molecular” viscosity. While the ansatz (VII.19) allows the rewriting of the Reynolds equation in the seemingly simpler form (VII.20)in which the two terms contributing to the effective pressure are to be seen as constituting a single field, it still involves an unknown, flow-dependent quantity, namely the effective viscosity νeff , which still needs to be specified. VII.22 Mixing-length model A first phenomenological hypothesis for the

turbulent viscosity is that implied in the mixinglength model (lxix) of Prandtl, which postulates the existence of a mixing length (lxx) `m , representing the typical scale over which momentum is transported by turbulence. The ansatz was motivated by an analogy with the kinetic theory of gases, in which the kinematic viscosity ν is proportional to the mean free path and to the typical velocity of particles. In practice, `m is determined empirically by the geometry of the flow. Under this assumption, the turbulent viscosity is given by νturb (t,~r) = `m (t,~r)2 ∂vx (t,~r) , ∂y (VII.22) in the case of a two-dimensional flow like the plane Couette flow (Sec. VI12), or for a more general motion νturb (t,~r) = `2m (t,~r) S (t,~r) , S| a typical value of the rate-of-shear tensor of the mean flow. In any case, the turbulent with |S viscosity is determined by local quantities. The latter point is actually a weakness of the model. For instance, it implies that the turbulent viscosity

(VII.22) vanishes at an extremum of the mean flow velocityfor instance, on the tube axis in the Hagen–Poiseuille flow, which is not realistic. In addition, turbulence can be transported from a region into another one, which also not describe by the ansatz. (lxix) Mischungswegansatz (lxx) Mischungsweglänge 118 Turbulence in non-relativistic fluids Eventually, the mixing-length model actually merely displaces the arbitrariness from the choice of the turbulent viscosity νturb to that of the mixing length `m , i.e it is just a change of unknown parameter. VII.23 k -model In order to describe the possible transport of turbulence within the mean flow, the so-called k-model was introduced. Denoting by k 0 ≡ 21~v02 the average kinetic energy of the turbulent fluctuations, the turbulent viscosity is postulated to be 1/2 νturb (t,~r) = `m (t,~r)k 0(t,~r) . (VII.23) An additional relation is needed to describe the transport of k 0 , to close the system of equations. For

simplicity, one the actual relation [see Eq (VII.26) below] is replaced by a similar-looking equation, in which the material derivative following the main flow of the average turbulent kinetic energy equals the sum of a transport termminus the gradient of a flux density, taken to be proportional to the gradient of k 0 , a production termnamely the energy extracted from the mean flow, and a dissipation term that describes the rate of energy release as heat, and whose form 3/2 Ė diss. = Ck0 /`m is motivated by dimensional arguments, with C a constant. Due to the introduction of this extra phenomenological transport equation for k 0 , which was not present in the mixing-length model, the k-model is referred to as a one-equation model .(lxxi) The k-model allows by construction the transport of turbulence. However, the mixing length `m remains an empirical parameter, and two further ones are introduced in the transport equation for the average turbulent kinetic energy. VII.24 (k

-ε)-model In the k-model, the dissipation term Ė diss. which stands for the ultimate transformation of turbulent kinetic energy into heat under the influence of viscous friction, and should thus be proportional to the viscosity ν, is determined by a dimensional argument. Another possibility is to consider the energy dissipation rate Ė diss (t,~r)which is usually rather denoted as ε̄as a dynamical variable, whose evolution is governed by a transport equation of its own. This approach yields a two-equation model ,(lxxii) the so-called (k-ε)-model A dimensional argument then gives `m ∼ k 0 3/2 /Ė diss. , and thus νturb (t,~r) = C k 0(t,~r) 2 , (VII.24) Ė diss. (t,~r) with C an empirical constant. In this modelor rather, this class of models, the mixing length is totally fixed by the dynamical variables, thus is no longer arbitrary. On the other hand, the two transport equations introduced for the average turbulent kinetic energy and the dissipation rate involve a

handful of parameters, which have to be determined empirically for each flow. In addition, the (k-ε)-model, like all descriptions involving a turbulent viscosity, relies on the assumption that the typical scale of variations of the mean flow velocity is clearly separated from the turbulent mixing length. This hypothesis is often not satisfied, in that many flows involve (lxxi) Eingleichungsmodell (lxxii) Zweigleichungsmodell VII.3 Statistical description of turbulence 119 turbulent motion over many length scales, in particular with a larger scale comparable with that of the gradients of the mean flow. In such flows, the notion of turbulent viscosity is not really meaningful. VII.3 Statistical description of turbulence Instead of handling the turbulent part of the motion like a source of momentum or a sink of kinetic energy for the mean flow, another approach consists in considering its dynamics more closely (Sec. VII31) As already argued in Sec VII14, this automatically

involves higher-order autocorrelation functions of the fluctuating velocity, which hints at the interest of looking at the general autocorrelation functions, rather than just their values at equal times and equal positions. This more general approach allows on the one hand to determine length scales of relevance for turbulence (Sec. VII32), and on the other hand to motivate a statistical theory of (isotropic) turbulence (Sec. VII33) VII.31 Dynamics of the turbulent motion Starting from the incompressible Navier–Stokes equation (VII.8) for the “total” flow velocity ~v and subtracting the Reynolds equation (VII.9) for the mean flow, one finds the dynamical equation governing the evolution of the turbulent velocity~v0, namely [for brevity, the (t,~r)-dependence of the fields is omitted]   0

0
∂~v ~ ~v − ∇ ~ P 0 + η4~v0 − ρ ~v0 · ∇ ~ ~ · ρ~v0 ⊗~v0 − T R . + ~v · ∇ ~v = −∇ ρ (VII.25a) ∂t or, after dividing by ρ and projecting along the xi -axis

of a coordinate system   0

∂ v0i ~ vi − d v0i v0j − v0i v0j . ~ v0i = − 1 dP + ν4v0i − ~v0 · ∇ + ~v · ∇ ∂t ρ dxi dxj (VII.25b) One recognizes in the left hand side of those equations the material derivative of the fluctuating velocity following the mean flow, D~v0/Dt. From the turbulent Navier–Stokes equation (VII.25), one finds for the average kinetic energy of the fluctuating motion k 0 ≡ 12~v02   X 3 3  3 3 X X Dk 0 d 1 0 0j X 1 0 0i 0j 0 0 ij 0 0i v0j S − 2ν =− P v + v v v − 2ν v S v S 0 ij S ij (VII.26) − ij i dxj ρ 2 i Dt j=1 i=1 i,j=1 i,j=1 with S 0 ij   2 ij ~ 0 1 dv0i dv0j + − g ∇ ·~v the components of the fluctuating rate-of-shear tensor. ≡ 2 dxj dxi 3 • The first term describes a turbulent yet conservative transportdue to pressure, convective transport by the fluctuating flow itself, or diffusive transport due the viscous friction, mixing the various length scales: the kinetic energy is transported without loss

from the large scales, comparable to that of the variations of the mean flow, to the smaller ones. This process is referred to as energy cascade. • The second term describes the “creation” of turbulent kinetic energy, which is actually extracted from the mean flow: it is preciselyup to the sign!the loss term in the Eq. (VII16) describing the transport of kinetic energy in the mean flow. • Eventually, the rightmost term in Eq. (VII26) represents the average energy dissipated as heat by the viscous friction forces, and will hereafter be denoted as Ė diss. 120 Turbulence in non-relativistic fluids In a statistically homogeneous and stationary turbulent flow, the amount of energy dissipated by viscous friction equals that extracted by turbulence from the mean flow, i.e − 3 X i,j=1 v0i v0j S ij = 2ν 3 X 0 S 0 ij S ij . (VII.27) i,j=1 VII.32 Characteristic length scales of turbulence VII.32 a Two-point autocorrelation function of the turbulent velocity

fluctuations :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: The fluctuations of the turbulent velocity~v0 are governed by an unknown probability distribution. Instead of knowing the latter, it is equivalent to rely on the (auto)correlation functions (n) κi1 i2 .in (t1 ,~r1 ; t2 ,~r2 ; ; tn ,~rn ) ≡ vi0 1(t1 ,~r1 ) vi0 2(t2 ,~r2 ) · · · vi0 n(tn ,~rn ) , in which the components of fluctuations at different instants and positions are correlated with each other. Remember that the 1-point averages vanish, Eq (VII4) The knowledge of all n-point autocorrelation functions is equivalent to that of the probability distribution. Yet the simplestboth from the experimental point of view as well as in numerical simulationsof these functions are the two-point autocorrelation functions [32] (2) κij (t,~r; t0 ,~r0 ) ≡ vi0 (t,~r) vj0 (t0 ,~r0 ) , (VII.28) which will hereafter be considered only at equal times t0 = t. In the case of a statistically

stationary turbulent flow,(41) the 2-point autocorrelation functions (2) κij (t,~r; t0 ,~r0 ) only depend on the time difference t0 − t, which vanishes if both instants are equal, yielding a function of ~r, ~r0 only. If the turbulence is in addition statistically homogeneous(41) which necessitates that one considers it far from any wall or obstacle, although this does not yet constitute a sufficient condition, then the 2-point autocorrelation function only depends on the separation ~ ≡ ~r0 − ~r of the two positions: X ~ . ~ = v0 (t,~r) v0 (t,~r + X) κij (X) i j (VII.29) If the turbulence is statistically locally isotropic,(41) the tensor κij only depends on the distance ~ between the two points. Such a statistical local isotropy often represents a good assumption X ≡ |X| for the structure of the turbulent motion on small scalesagain, far from the boundaries of the flowand will be assumed hereafter. ~ Let ~ek denote a unit vector along X, ~ ~e⊥ a unit vector in Consider

two points at ~r and ~r + X. 0 a direction orthogonal to ~ek , and ~e⊥ perpendicular to both ~ek and ~e⊥ . The component vk0 of the ~ turbulent velocityat ~r or ~r + Xalong ~ek is referred to as “longitudinal”, those along ~e⊥ or ~e⊥0 0 0 (v⊥ , v⊥0 ) as “lateral”. The autocorrelation function (VII.29) can be expressed with the help of the two-point functions ~ and κ0 (X) ≡ v0 (t,~r) v0 0 (t,~r + X) ~ as ~ κ⊥ (X) ≡ v0 (t,~r) v0 (t,~r + X), κk(X) ≡ vk0 (t,~r) vk0 (t,~r + X), ⊥ ⊥ ⊥ ⊥ ⊥ 3 X  Xi Xj  ijk Xk 0 κij (X) = , κk(X) − κ⊥ (X) + κ⊥ (X) δij + κ⊥ (X) ~2 X X k=1 ~ where the last term vanishes for statistically space-parity with {Xi } the Cartesian components of X, (42) invariant turbulence, which is assumed to be the case from now on.(43) This means that the probability distribution of the velocity fluctuations ~v0 is stationary (time-independent) resp. homogeneous (position-independent) resp. locally isotropic (the same for

all Cartesian components of ~v0) (42) Invariance under the space-parity operation is sometimes considered to be part of the isotropy, sometimes not. (43) In presence of a magnetic fieldi.e in the realm of magnetohydrodynamics, this last term is indeed present (41) 121 VII.3 Statistical description of turbulence ~ ·~v0 = 0 with vj and averaging yields Multiplying the incompressibility condition ∇ 3 X ∂κij (X) i=1 ∂Xi = 0, resulting in the identity X dκk(X) , 2 dX which means that κij can be expressed in terms of the autocorrelationfunction κk only. κ⊥ (X) = κk(X) + VII.32 b Microscopic and macroscopic length scales of turbulence :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: The assumed statistical isotropy gives κk (0) = [vk(t,~r)]2 = 31 [~v0(t,~r)]2 : let f (X) be the function such that κk (X) ≡ 31 [~v0(t,~r)]2 f (X) and that • f (0) = 1; • the fluctuations of the velocity at points separated by a large distance X are not

correlated with another, so that κk(X) must vanish: lim f (X) = 0. X∞ • In addition, f is assumed to be integrable over R+ , and such that its integral from 0 to +∞ is convergent. The function f then defines a typical macroscopic length scale, namely that over which f resp. κk decreases,(44) the integral scale or external scale (lxxiii) Z ∞ LI ≡ f (X) dX. (VII.30) 0 Empirically, this integral scale is found to be comparable to the scale of the variations of the mean flow velocity, i.e characteristic for the production of turbulence in the flow For example, in a flow past an obstacle, LI is of the same order of magnitude as the size of the obstacle. Assumingas has been done till nowlocally isotropic and space-parity invariant turbulence, the function f (X) is even, so that its Taylor expansion around X = 0 defines a microscopic length scale:   1 X 2 1 f (X) 1 − + O(X 4 ) with `2T ≡ − 00 > 0. (VII.31) X0 2 `T f (0) `T is the Taylor microscale.(lxxiv)(45) ~ One

finds Let xk denote the coordinate along X. 2 . 0 2  `2T = vk0 (t,~r) dvk(t,~r)/dxk , (VII.32) i.e `T is the typical length scale of the gradients of the velocity fluctuations Using the definition of f , the Taylor expansion (VII.31) can be rewritten as ~ vk0 (t,~r) vk0 (t,~r + X) [vk0 (t,~r)]2 1+ X0 1 vk0 (t,~r) ∂k2 vk0 (t,~r) 2 X , 2 [vk0 (t,~r)]2 where ∂k denotes the derivative with respect to xk . Invoking the statistical homogeneity of the turbulence, [vk0 (t,~r)]2 is independent of position, thus of xk , which after differentiation leads successively to vk0 (t,~r) ∂kvk0 (t,~r) = 0 and then [∂kvk0 (t,~r)]2 +vk0 ∂k2 vk0 (t,~r) = 0, proving relation (VII.32) −X/LI The reader should think of the example κk(X) = κk(0) e−X/LI , or at least κk(X) ∝ for X large enough ∼e compared to a microscopic scale much smaller than LI . (45) . named after the fluid dynamics practitioner G I Taylor, not after B Taylor of the Taylor series (44) (lxxiii)

Integralskala, äußere Skala (lxxiv) Taylor-Mikroskala 122 Turbulence in non-relativistic fluids Remark: Even if the Taylor microscale emerges naturally from the formalism, it does not represent the length scale of the smallest eddies in the flow, despite what one could expect. To find another, physically more relevant microscopic scale, it is necessary to investigate the behavior of the longitudinal increment ~ − vk0 (t,~r) δvk0 (X) ≡ vk0 (t,~r + X) (VII.33) of the velocity fluctuations, which compares the values of the longitudinal component of the latter at different points. According to the definition of the derivative, dvk0 /dxk is the limit when X 0 of the ratio δvk0 (X)/X. The microscopic Kolmogorov length scale `K is then defined by  0  dvk(t,~r) 2 [δvk0 (`K )]2 [δvk0 (X)]2 ≡ lim . (VII.34) = X0 X2 dxk `2K The role of this length scale will be discussed in the following Section, yet it can already be mentioned that it is the typical scale of the

smallest turbulent eddies, thus the pendant to the integral scale LI . Remark: Squaring the longitudinal velocity increment (VII.33) and averaging under consideration of the statistical homogeneity, one finds when invoking Eq. (VII31)   [δvk0 (X)]2 1 X 2 . ∼ 2[vk0 (X)]2 X0 2 `T On the other hand, experiments or numerical simulations show that the left hand side of this relation equals about 1 when X is larger than the integral scale LI . That is, the latter and the Taylor microscale can also be recovered from the longitudinal velocity increment. VII.33 The Kolmogorov theory (K41) of isotropic turbulence A first successful statistical theory of turbulence was proposed in 1941 by Kolmogorov(ao) for statistically locally isotropic turbulent motion, assuming further stationarity, homogeneity and spaceparity invariance [33, 34]. This K41-theory describes the fluctuations of the velocity increments δvi0 (X), and relies on two assumptionsoriginally termed similarity hypotheses by

Kolmogorov: 1st Kolmogorov hypothesis The probability distributions of the turbulent-velocity increments δvi0 (X), i = 1, 2, 3, are universal on separation scales X small compared to the integral scale LI , and are entirely determined by the kinematic viscosity ν of the fluid and by the average energy dissipation rate per unit mass Ė diss. (K41-1) Here “universality” refers to an independence from the precise process which triggers the turbulence. Considering e.g the longitudinal increment, this hypothesis gives for the second moment of the probability distribution q    3 1/4 ν (2) X 0 2 [δvk(X)] = ν Ė diss. Φ for X  LI with `K = (VII.35) `K Ė diss. q and Φ(2) a universal function, irrespective of the flow under study. The factor ν Ė diss and the form of `K follow from dimensional considerationsthe
n-point autocorrelation function n/4 involves another function Φ(n) multiplying a factor ν Ė diss. . (ao) A. N Kolmogorov = A. N Kolmogorov, 1903–1987

123 VII.3 Statistical description of turbulence The hypothesis (K41-1) amounts to assuming that the physics of the fluctuating motion, far from the scale at which turbulence is created, is fully governed by the available energy extracted from the mean flowwhich in the stationary regime equals the average energy dissipated by viscous friction in the turbulent motionand by the amount of friction. 2nd Kolmogorov hypothesis The probability distributions of the turbulent-velocity increments δvi0 (X), i = 1, 2, 3, is independent of the kinematic viscosity ν of the fluid on separation scales X large compared to the microscopic scale `K . (K41-2) The idea here is that viscous friction only plays a role at the microscopic scale, while the rest of the turbulent energy cascade is conservative. The assumption holds for the longitudinal increment (VII.35) if and only if Φ(2)(x) ∼ B (2) x2/3 x1 with B (2) a universal constant, i.e if
2/3 for `K  X  LI . (VII.36) [δvk0 (X)]2 ∼ B (2)

Ė diss. X The Kolmogorov 2/3-law (VII.36) does not involve any length scale: this reflects the lengthscale “self-similarity” of the conservative energy-cascading process in the inertial range (lxxv) `K  X  LI , in which the only relevant parameter is the energy dissipation rate. The increase of the autocorrelation function [δvk0 (X)]2 as X 2/3 is observed both experimentally and in numerical simulations.(46) A further prediction of the K41-theory regards the energy spectrum of the turbulent motion. Let ~v˜ 0(t, ~k) denote the spatial Fourier transform of the fluctuating velocity. Up to a factor involving the inverse of the (infinite) volume of the flow, the kinetic energy per unit mass of the turbulent motion component with wave vector equal to ~k up to d3~k is 21 [~v˜ 0(t, ~k)]2 d3~k. In the case of statistically isotropic turbulence, 21 [~v˜ 0(t, ~k)]2 d3~k = 2πk 2 [~v˜ 0(t, ~k)]2 dk ≡ SE (k) dk with SE (k) the kinetic-energy spectral density. From the 2/3-law

(VII.36),(47) one can then derive the −5/3-law for the latter, namely  1/4 2/3 ε̄ −1 , (VII.37) SE (k) = CK Ė diss. k −5/3 for L−1  k  ` = I K ν3 with CK a universal constant, the Kolmogorov constant, independent from the fluid or the flow geometry, yet dependinglike the −5/3-law itselfon the space dimensionality. Experimentally(46) one finds CK ≈ 1.45 As already mentioned, the laws (VII.36) and (VII37) provide a rather satisfactory description of the results of experiments or numerical simulations. The K41-theory also predicts that the higher-order moments of the probability distribution of the velocity increments should be universal
n/3 as welland the reader can easily determine their scaling behavior [δvk0 (X)]n ∼ B (n) Ė diss. X in the inertial range using dimensional arguments, yet this prediction is no longer supported by experiment: the moments do depend on X as power laws, yet not with the predicted exponents. A deficiency of Kolmogorov’s theory is

that in his energy cascade, only eddies of similar size interact with each other to transfer the energy from large to small length scales, which is encoded in the self-similarity assumption. In that picture, the distribution of the eddy sizes is statistically stationary. (46) (47) Examples from experimental results are presented in Ref. [35, Chapter 5] . and assuming that SE (k) behaves properly, ie decreases quickly enough, at large k (lxxv) Trägheitsbereich 124 Turbulence in non-relativistic fluids In contrast, turbulent motion itself tends to deform eddies, by stretching vortices into tubes of smaller cross section, until they become so small that shear viscosity becomes efficient to counteract this process (see Sec. VI5) This behavior somewhat clashes with Kolmogorov’s picture. Bibliography for Chapter VII • Chandrasekhar [36]. • Feynman [8, 9] Chapter 41-4–41–6. • Faber [1] Chapter 9.1, 92–96 • Frisch [35]. • Guyon et al. [2] Chapter 12 •

Landau–Lifshitz [3, 4] Chapter III § 33–34. C HAPTER VIII Convective heat transfer The previous two Chapters were devoted to flows dominated by viscosity (Chap. VI) or by convective motion (Chap. VII) In either case, the energy-conservation equation (III35), and in particular the term representing heat conduction, was never taken into account, with the exception of a brief mention in the study of static Newtonian fluids (Sec. VI11) The purpose of this Chapter is to shift the focus, and to discuss motions of Newtonian fluids in which heat is transfered from one region of the fluid to another. A first such type of transfer is heat conduction, which was already encountered in the static case. Under the generic term “convection”, or “convective heat transfer”, one encompasses flows in which heat is also transported by the moving fluid, not only conductively. Heat transfer will be caused by differences in temperature in a fluid. Going back to the equations of motion, one

can make a few assumptions so as to eliminate or at least suppress other effects, and emphasize the role of temperature gradients in moving fluids (Sec VIII.1) A specific instance of fluid motion driven by a temperature difference, yet also controlled by the fluid viscosity, which allows for a richer phenomenology, is then presented in Sec. VIII2 VIII.1 Equations of convective heat transfer The fundamental equations of the dynamics of Newtonian fluids seen in Chap. III include heat conduction, in the form of a term involving the gradient of temperature, yet the change in time of temperature does not explicitly appear. To obtain an equation involving the time derivative of temperature, some rewriting of the basic equations is thus needed, which will be done together with a few simplifications (Sec. VIII11) Conduction in a static fluid is then recovered as a limiting case. In many instances, the main effect of temperature differences is however rather to lead to variations of the mass

density, which in turn trigger the fluid motion. To have a more adapted description of such phenomena, a few extra simplifying assumptions are made, leading to a new, closed set of coupled equations (Sec. VIII12) VIII.11 Basic equations of heat transfer ~ Consider a Newtonian fluid submitted to conservative volume forces f~V = −ρ∇Φ. Its motion is governed by the laws established in Chap. III, namely by the continuity equation, the Navier–Stokes equation, and the energy-conservation equation or equivalently the entropy-balance equation, which we now recall. Expanding the divergence of the mass flux density, the continuity equation (III.9) becomes Dρ(t,~r) ~ ·~v(t,~r). = −ρ(t,~r)∇ Dt (VIII.1a) In turn, the Navier–Stokes equation (III.30a) may be written in the form ρ(t,~r)     D~v(t,~r) ~ P (t,~r) − ρ(t,~r)∇Φ(t,~ ~ ~ · η(t,~r)S ~ ζ(t,~r)∇ ~ ·~v(t,~r) . (VIII1b) S(t,~r) + ∇ = −∇ r) + 2∇ Dt 126 Convective heat transfer Eventually,

straightforward algebra using the continuity equation allows one to rewrite the entropy balance equation (III.40b) as       D s(t,~r) ~ ·~v(t,~r) 2 . (VIII1c) ~ · κ(t,~r)∇T ~ (t,~r) + 2η(t,~r) S (t,~r) : S (t,~r) + ζ(t,~r) ∇ ρ(t,~r) =∇ Dt ρ(t,~r) T (t,~r) T (t,~r) Since we wish to isolate effects directly related with the transfer of heat, or playing a role in it, we shall make a few assumptions, so as to simplify the above set of equations. • The transport coefficients η, ζ, κ depend on the local thermodynamic state of the fluid, i.e on its local mass density ρ and temperature T , and thereby indirectly on time and position. Nevertheless, they will be taken as constant and uniform throughout the fluid, and taken out of the various derivatives in Eqs. (VIII1b)–(VIII1c) This is a reasonable assumption as long as only small variations of the fluid properties are considered, which is consistent with the next assumption. Somewhat abusively, we shall in fact even

allow ourselves to consider η resp. κ as uniform in Eq. (VIII1b) resp (VIII1c), later replace them by related (diffusion) coefficients ν = η/ρ resp. α = κ/ρc P , and then consider the latter as uniform constant quantities The whole procedure is only “justified” in that one can checkby comparing calculations using this assumption with numerical computations performed without the simplifications that it does not lead to omitting a physical phenomenon. • The fluid motions under consideration will be assumed to be “slow”, i.e to involve a small flow velocity, in the following sense: ~ v(t,~r) = 0 will hold on the right hand sides of each of – The incompressibility condition ∇·~ Eqs. (VIII1) Accordingly, Eq (VIII1a) simplifies to Dρ(t,~r)/Dt = 0 while Eq (VIII1b) becomes the incompressible Navier–Stokes equation  ∂~v(t,~r)  ~ ~v(t,~r) = − 1 ∇ ~ P (t,~r) − ∇Φ(t,~ ~ + ~v(t,~r) · ∇ r) + ν4~v(t,~r), (VIII.2) ∂t ρ(t,~r) in which the kinematic

viscosity ν is taken to be constant. – The rate of shear is small, so that its square can be neglected in Eq. (VIII1c) Accordingly, that equation simplifies to   D s(t,~r) ρ(t,~r) = κ4T (t,~r). (VIII.3) Dt ρ(t,~r) The left member of that equation can be further rewritten. Dividing the fundamental relation of thermodynamics dU = T dS − P dV (at constant particle number) by the mass of the atoms of the fluid yields the relation       e s 1 d =Td −P d . ρ ρ ρ In keeping with the assumed incompressibility of the motion, the rightmost term vanishes, while the change in specific energy can be related to the variation of temperature as d(e/ρ) = c P dT with c P the specific heat capacity at constant pressure. In a fluid particle, one may thus write   s Td = c P dT, (VIII.4) ρ which translates into a relation between material derivatives when the fluid particles are followed in their motion. The left member of Eq (VIII3) may then be expressed in terms of the substantial

derivative of the temperature. Introducing the thermal diffusivity (lxxvi) κ , (VIII.5) α≡ ρc P (lxxvi) Temperaturleitfähigkeit 127 VIII.1 Equations of convective heat transfer which will be assumed to be constant and uniform in the fluid, where ρc P is the volumetric heat capacity at constant pressure, one eventually obtains  DT (t,~r) ∂T (t,~r)  ~ T (t,~r) = α4T (t,~r) = + ~v(t,~r) · ∇ Dt ∂t (VIII.6) which is sometimes referred to as (convective) heat transfer equation. ~ If the fluid is at rest or if its velocity is “small” enough that the convective part ~v · ∇T be negligible, Eq. (VIII6) simplifies to the classical heat diffusion equation, with diffusion constant α The thermal diffusivity α thus measures the ability of a medium to transfer heat diffusively, just like the kinematic shear viscosity ν quantifies the diffusive transfer of momentum. Accordingly, both have the same dimension L2 T−1 , and their relative strength can be measured by

the dimensionless Prandtl number ν ηc P Pr ≡ = (VIII.7) α κ which in contrast to the Mach, Reynolds, Froude, Ekman, Rossby. numbers encountered in the previous Chapters is entirely determined by the fluid, independent of any flow characteristics. VIII.12 Boussinesq approximation If there is a temperature gradient in a fluid, it will lead to a heat flux density, and thereby to a transfer of heat, thus influencing the fluid motion. However, heat exchanges by conduction are often slowexcept in metals, so that another effect due to temperature differences is often the first to play a significant role, namely thermal expansion (or contraction), which will lead to buoyancy (Sec. IV14) when a fluid particle acquires a mass density different from that of its surroundings The simplest approach to account for this effect, due to Boussinesq,(48) consists in considering that even though the fluid mass density changes, nevertheless the motion can be to a very good approximation viewed as

incompressiblewhich is what was assumed in Sec. VIII11: ~ ·~v(t,~r) 0, ∇ (VIII.8) where is used to allow for small relative variations in the mass density, which is directly related to the expansion rate [Eq. (VIII1a)] Denoting by T0 a typical temperature in the fluid and ρ0 the corresponding mass density (strictly speaking, at a given pressure), the effect of thermal expansion on the latter reads ρ(Θ) = ρ0 (1 − α(V ) Θ), (VIII.9) Θ ≡ T − T0 (VIII.10) with the temperature difference measured with respect to the reference value, and   1 ∂ρ α(V ) ≡ − ρ ∂T P ,N (VIII.11) the thermal expansion coefficient for volume, where the derivative is taken at the thermodynamic point corresponding to the reference value ρ0 . Strictly speaking, the linear regime (VIII9) only holds when α(V ) Θ  1, which will be assumed hereafter. (48) Hence its denomination Boussinesq approximation (for buoyancy). 128 Convective heat transfer Consistent with relation

(VIII.9), the pressure term in the incompressible Navier–Stokes equation can be approximated as ~ P (t,~r)   1 ~ ∇ 1 + α(V ) Θ(t,~r) . − ∇P (t,~r) − ρ(t,~r) ρ0 Introducing an effective pressure P eff which accounts for the leading effect of the potential from which the volume forces derive, P eff.(t,~r) ≡ P (t,~r) + ρ0 Φ(t,~r), one finds ~ P eff.(t,~r) 1 ~ ∇ ~ ~ ∇P (t,~r) − ∇Φ(t,~ r) − + α(V ) Θ(t,~r)∇Φ(t,~ r), ρ(t,~r) ρ0 ~ P eff. has been dropped To this level of approximation, where a term of subleading order α(V ) Θ∇ the incompressible Navier–Stokes equation (VIII.2) becomes − ~  ∂~v(t,~r)  ~ ~ ~v(t,~r) = − ∇P eff.(t,~r) + α(V ) Θ(t,~r)∇Φ(t,~ r) + ν4~v(t,~r). (VIII.12) + ~v(t,~r) · ∇ ∂t ρ0 This form of the Navier–Stokes equation emphasizes the role of a finite temperature difference Θ in providing an extra force density which contributes to the buoyancy, supplementing the effective pressure term. Eventually,

definition (VIII.10) together with the convective heat transfer equation (VIII6) lead at once to  ∂Θ(t,~r)  ~ Θ(t,~r) = α4Θ(t,~r). + ~v(t,~r) · ∇ (VIII.13) ∂t The (Oberbeck (ap) –)Boussinesq equations (VIII.8), (VIII12), and (VIII13) represent a closed system of five coupled scalar equations for the dynamical fields~v, Θwhich in turn yields the whole variation of the mass densityand P eff. VIII.2 Rayleigh–Bénard convection A relatively simple example of flow in which thermal effects play a major role is that of a fluid between two horizontal plates at constant but different temperatures, the lower plate being at the ~ higher temperature, in a uniform gravitational potential −∇Φ(t,~ r) = ~g , in the absence of horizontal pressure gradient. The distance between the two plates will be denoted by d, and the temperature difference between them by ∆T , where ∆T > 0 when the lower plate is warmer. When needed, a system of Cartesian coordinates will be used,

with the (x, y)-plane midway between the plates and a vertical z-axis, with the acceleration of gravity pointing towards negative values of z. VIII.21 Phenomenology of the Rayleigh–Bénard convection VIII.21 a Experimental findings :::::::::::::::::::::::::::::::: If both plates are at the same temperature or if the upper one is the warmer (∆T < 0), the fluid between them can simply be at rest, with a stationary linear temperature profile. As a matter of fact, denoting by T0 resp. P 0 the temperature resp pressure at a point at z = 0 and ρ0 the corresponding mass density, one easily checks that equations (VIII.8), (VIII12), (VIII13) admit the static solution z2 z ~vst.(t,~r) = ~0, Θst(t,~r) = − ∆T, P eff,st(t,~r) = P 0 − ρ0 g α(V ) ∆T, (VIII.14) d 2d (ap) A. Oberbeck, 1849–1900 129 VIII.2 Rayleigh–Bénard convection with the pressure given by P st.(t,~r) = P eff,st(t,~r) − ρ0 gz Since |z/d| < 12 and α(V ) ∆T  1, one sees that the main part of

the pressure variation due to gravity is already absorbed in the definition of the effective pressure. If ∆T = 0, one recognizes the usual linear pressure profile of a static fluid at constant temperature in a uniform gravity field. One can check that the fluid state defined by the profile (VIII.14) is stable against small perturbations of any of the dynamical fields To account for that property, that state (for a given temperature difference ∆T ) will be referred to as “equilibrium state”. Increasing now the temperature of the lower plate with respect to that of the upper plate, for small positive temperature differences ∆T nothing happens, and the static solution (VIII.14) still holdsand is still stable. When ∆T reaches a critical value ∆Tc , the fluid starts developing a pattern of somewhat regular cylindrical domains rotating around their longitudinal, horizontal axes, two neighboring regions rotating in opposite directions. These domains in which warmer and thus less

dense fluid rises on the one side while colder, denser fluid descends on the other side, are called Bénard cells.(aq) 6 d ? Figure VIII.1 – Schematic representation of Bénard cells between two horizontal plates The transition from a situation in which the static fluid is a stable state, to that in which motion developsi.e the static case is no longer stable, is referred to as (onset of the) Rayleigh–Bénard instability. Since the motion of the fluid appears spontaneously, without the need to impose any external pressure gradient, it is an instance of free convection or natural convectionin opposition to forced convection). Remarks: ∗ Such convection cells are omnipresent in Nature, as e.g in the Earth mantle, in the Earth atmosphere, or in the Sun convective zone. ∗ When ∆T further increases, the structure of the convection pattern becomes more complicated, eventually becoming chaotic. In a series of experiments with liquid helium or mercury, A. Libchaber(ar) and his

collaborators observed the following features [37, 38, 39]: Shortly above ∆Tc , the stable fluid state involve cylindrical convective cells with a constant profile. Above a second threshold, “oscillatory convection” develops: that is, undulatory waves start to propagate along the “surface” of the convective cells, at first at a unique (angular) frequency ω1 , thenas ∆T further increasesalso at higher harmonics n1 ω1 , n1 ∈ N. As the temperature difference ∆T reaches a third threshold, a second undulation frequency ω2 appears, incommensurate with ω1 , later accompanied by the combinations n1 ω1 + n2 ω2 , with n1 , n2 ∈ N. At higher ∆T , the oscillator with frequency ω2 experiences a shift from its proper frequency to a neighboring submultiple of ω1 e.g , ω1 /2 in the experiments with He, illustrating the phenomenon of frequency locking. For even higher ∆T , submultiples of ω1 appear (“frequency demultiplication”), then a low-frequency continuum, and

eventually chaos. (aq) H. Bénard, 1874–1939 (ar) A. Libchaber, born 1934 130 Convective heat transfer Each appearance of a new frequency may be seen as a bifurcation. The ratios of the experimentally measured lengths of consecutive intervals between successive bifurcations provide an estimate of the (first) Feigenbaum constant (as) in agreement with its theoretical valuethereby providing the first empirical confirmation of Feigenbaum’s theory. VIII.21 b Qualitative discussion :::::::::::::::::::::::::::::::: Consider the fluid in its “equilibrium” state of rest, in the presence of a positive temperature difference ∆T , so that the lower layers of the fluid are warmer than the upper ones. If a fluid particle at altitude z acquires, for some reason, a temperature that differs from the equilibrium temperaturemeasured with respect to some reference valueΘ(z), then its mass density given by Eq. (VIII9) will differ from that of its environment As a result, the

Archimedes force acting on it no longer exactly balances its weight, so that it will experience a buoyancy force. For instance, if the fluid particle is warmer that its surroundings, it will be less dense and experience a force directed upwards. Consequently, the fluid particle should start to move in that direction, in which case it encounters fluid which is even colder and denser, resulting in an increased buoyancy and a continued motion. According to that reasoning, any vertical temperature gradient should result in a convective motion. There are however two effects that counteract the action of buoyancy, and explain why the Rayleigh–Bénard instability necessitates a temperature difference larger than a given threshold. First, the rising particle fluid will also experience a viscous friction force from the other fluid regions it passes through, which slows its motion. Secondly, if the rise of the particle is too slow, heat has time to diffuseby heat conductionthrough its

surface: this tends to equilibrate the temperature of the fluid particle with that of its surroundings, thereby suppressing the buoyancy. Accordingly, we can expect to find that the Rayleigh–Bénard instability will be facilitated when α(V ) ∆T gi.e the buoyancy per unit massincreases, as well as when the thermal diffusivity α and the shear viscosity ν decrease. Translating the previous argumentation in formulas, let us consider a spherical fluid particle with radius R, and assume that it has some vertically directed velocity v, while its temperature initially equals that of its surroundings. With the fluid particle surface area, proportional to R2 , and the thermal diffusivity κ, one can estimate the characteristic time for heat exchanges between the particle and the neighboring fluid, namely R2 τQ = C α with C a geometrical factor. If the fluid particle moves with constant velocity v during that duration τQ , while staying at almost constant temperature since heat

exchanges remain limited, the temperature difference δΘ it acquires with respect to the neighboring fluid is δΘ = ∂Θ ∂Θ ∆T R2 δz = vτQ = C v, ∂z ∂z d α where ∆T /d is the temperature gradient imposed by the two plates in the fluid. This temperature difference gives rise to a mass density difference δρ = −ρ0 α(V ) δΘ = −Cρ0 v R2 α(V ) ∆T , α d between the particle and its surroundings. As a result, fluid particle experiences an upwards directed buoyancy 4π 3 4πC R5 α(V ) ∆T − R δρg = ρ0 gv . (VIII.15) 3 3 α d (as) M. Feigenbaum, born 1944 131 VIII.2 Rayleigh–Bénard convection On the other hand, the fluid particle is slowed in its vertical motion by the downwards oriented Stokes friction force acting on it, namely, in projection on the z-axis FStokes = −6πRηv. (VIII.16) Note that assuming that the velocity v remains constant, with a counteracting Stokes force that is automatically the “good” one, relies on the picture

that viscous effects adapt instantaneously, i.e that momentum diffusion is fast. That is, the above reasoning actually assumes that the Prandtl number (VIII.7) is much larger than 1; yet its result is independent from that assumption Comparing Eqs. (VIII15) and (VIII16), buoyancy will overcome friction, and thus the Rayleigh– Bénard instability take place, when α(V ) ∆T gR4 4πC R5 α(V ) ∆T 9 ρ0 gv > 6πRρ0 νv ⇔ > . 3 α d ανd 2C Note that the velocity v which was invoked in the reasoning actually drops out from this condition. Taking for instance R = d/2which maximizes the left member of the inequality, this becomes α(V ) ∆T g d3 72 > = Rac . να C Ra is the so-called Rayleigh number and Rac its critical value, above which the static-fluid state is instable against perturbation and convection takes place. The “value” 72/C found with the above simple reasoning on force equilibrium is totally irrelevantboth careful experiments and theoretical

calculations agree with Rac = 1708 for a fluid between two very large plates, the important lesson is the existence of a threshold. Ra ≡ VIII.22 Toy model for the Rayleigh–Bénard instability A more refinedalthough still crudetoy model of the transition to convection consists in considering small perturbations ~v, δΘ, δ P eff. around a static state ~vst = ~0, Θst , P eff,st , and to linearize the Boussinesq equations to first order in these perturbations. As shown by Eq (VIII14), the effective pressure P eff.,st actually already includes a small correction, due to α(V ) ∆T being much smaller than 1, so that we may from the start neglect δ P eff. To first order in the perturbations, Eqs. (VIII12), projected on the z-axis, and (VIII13) give, after subtraction of the contributions from the static solution ∂vz = ν4vz + α(V ) δΘg, (VIII.17a) ∂t ∂δΘ ∆T − vz = α4δΘ. (VIII.17b) ∂t d Moving the second term of the latter equation to the right hand side

increases the parallelism of this set of coupled equations. In addition, there is also the projection of Eq (VIII12) along the x-axis, and the velocity field must obey the incompressibility condition (VIII.8) The proper approach would now be to specify the boundary conditions, namely: the vanishing of vz at both platesimpermeability condition, the vanishing of vx at both platesno-slip condition, and the identity of the fluid temperature at each plate with that of the corresponding plate; that is, all in all, 6 conditions. By manipulating the set of equations, it can be turned into a 6th-order linear partial differential equation for δΘ, on which the boundary conditions can be imposed. Instead of following this long road,(49) we refrain from trying to really solve the equations, but rather make a simple ansatz, namely vz (t,~r) = v0 eγt cos(kx)which automatically fulfills the (49) The reader may find details in Ref. [40, Chap II] 132 Convective heat transfer incompressibility

equation, but clearly violates the impermeability conditions, and a similar one for δΘ, with γ a constant. Substituting these forms in Eqs (VIII17) yield the linear system
γv0 = −k 2 νv0 + α(V ) δΘ0 g ⇔ γ + νk 2 v0 − gα(V ) δΘ0 = 0,
∆T ∆T γδΘ0 = −k 2 αδΘ0 + v0 ⇔ v0 − γ + αk 2 δΘ0 = 0 d d for the amplitudes v0 , δΘ0 . This admits a non-trivial solution only if

α(V ) ∆T γ + νk 2 γ + αk 2 − g = 0. (VIII.18) d This is a straightforward quadratic equation for γ. It always has two real solutions, one of which is negativecorresponding to a dampened perturbationsince their sum is −(α + ν)k 2 < 0; the other solution may change sign since their product α(V ) ∆T ανk 4 − g d is positive for ∆T = 0, yielding a second negative solution, yet changes sign as ∆T increases. The vanishing of this product thus signals the onset of instability. Taking for instance k = π/d to fix ideas, this occurs at a critical Rayleigh number

Rac = α(V ) ∆T g d3 = π4, αν where the precise value (here π 4 ) is irrelevant. From Eq. (VIII18) also follows that the growth rate of the instability is given in the neighborhood of the threshold by Ra − Rac αν 2 k , γ= Rac α + ν i.e it is infinitely slow at Rac This is reminiscent of a similar behavior in the vicinity of the critical point associated with a thermodynamic phase transition. By performing a more rigorous calculation including non-linear effects, one can show that the velocity amplitude at a given point behaves like  β Ra − Rac 1 (VIII.19) v∝ with β = Rac 2 in the vicinity of the critical value, and this prediction is borne out by experiments [41]. Such a power law behavior is again reminiscent of the thermodynamics of phase transitions, more specifically heresince v vanishes below Rac and is finite aboveof the behavior of the order parameter in the vicinity of a critical point. Accordingly the notation β used for the exponent in relation

(VIII.19) is the traditional choice for the critical exponent associated with the order parameter of phase transitions. Eventually, a last analogy with phase transitions regards the breaking of a symmetry at the threshold for the Rayleigh–Bénard instability. Below Rac , the system is invariant under translations parallel to the plates, while above Rac that symmetry is spontaneously broken. Bibliography for Chapter VIII • A nice introduction to the topic is to be found in Ref. [42], which is a popular science account of part of Ref. [43] • Faber [1] Chapter 8.5–87 & 92 • Guyon et al. [2] Chapter 112 • Landau–Lifshitz [3, 4] Chapter V § 49–53 & 56–57. C HAPTER IX Fundamental equations of relativistic fluid dynamics When the energy density becomes largeas may happen for instance in compact astrophysical objects, in the early Universe, or in high-energy collisions of heavy nucleithe “atoms” constituting a fluid can acquire very high kinetic energies,

that become comparable to their (rest) mass energy. A non-relativistic description of the medium is then no longer adapted, and must be replaced by a relativistic model. Some introductory elements of such a description are presented in this Chapter in which the basic laws governing the dynamics of relativistic fluids are formulated and discussed, and the following onewhich will deal with a few simple analytically tractable solutions of the equations. As in the non-relativistic case, the basic equations governing the motion of a fluid in the relativistic regime are nothing but formulations of the most fundamental laws of physics, namely conservation laws for “particle number”in fact, for the conserved quantum numbers carried by particles, and for energy and momentum (Sec. IX1) Precisely because the equations simply express general conservation laws, they are not very specific, and contain at first too many degrees of freedom to be tractable. To make progress, it is necessary to

introduce models for the fluid under consideration, leading for instance to distinguishing between perfect and dissipative fluids. A convenient way to specify the constitutive equations characteristic of such models is to do so in terms of a fluid four-velocity, which generalizes the non-relativistic flow velocity, yet in a non-unique way (Sec. IX2) Such a fluid four-velocity also automatically singles out a particular reference frame, the local rest frame, in which the conserved currents describing the physics of the fluid take a simpler form, whose physical interpretation is clearer. The perfect fluids are thus those whose properties at each point are spatially isotropic in the corresponding local rest frame, from which there follows that the conserved currents can only depend on the flow four-velocity, not on its derivatives (Sec. IX3) Conversely, when the conserved currents involve (spatial) gradients of the fluid four-velocity, these derivatives signal (real) fluids with

dissipative effects (Sec. IX4) Two topics that lie beyond the main stream of this Chapter are given in appendices, namely the expression of the conserved currents of relativistic fluid dynamics in terms of underlying microscopic quantities (Sec. IXA) and a discussion of relativistic kinematics (Sec IXB) Throughout this Chapter and the next one, the fluids occupy domains of the four-dimensional Minkowski space-time M 4 of Special Relativity. The position of a generic point of M 4 will be designated by a 4-vector x. Given a reference frame R and a system of coordinates, those of x will be denoted by {xµ } ≡ (x0 , x1 , x2 , x3 )where in the case of Minkowski coordinates(50) x0 = ct with t the time measured by an observer at rest in R. (50) We shall call Minkowski coordinates the analog on the space-time M 4 of the Cartesian coordinates on Euclidean space E 3 , i.e those corresponding to a set of four mutually orthogonal 4-vectors (e0 , e1 , e2 , e3 ) such that the metric tensor has

components gµν = eµ · eν = diag(−1, +1, +1, +1) for µ, ν = 0, 1, 2, 3. They are also alternatively referred to as Lorentz coordinates. 134 Fundamental equations of relativistic fluid dynamics For the metric tensor g on M 4 , we use the “mostly plus” convention, with signature (−, +, +, +), i.e in the case of Minkowski coordinates x0 = −x0 while xi = xi for i = 1, 2, 3 Thus, time-like resp. space-like 4-vectors have a negative resp positive semi-norm IX.1 Conservation laws As stated in the introduction, the equations governing the dynamics of fluids in the relativistic, just as in the non-relativistic case, embody conservation principles. More precisely, they are differential formulations of these laws. Instead of proceeding as in Chap III, in which the local formulations were derived from integral ones, we shall hereafter postulate the differential conservation laws, and check or argue that they lead to the expected macroscopic behavior. Starting from the local

level is more natural here, since one of the tenets underlying relativistic theories, as e.g quantum field theory, is precisely localitythe absence of action at distance besides causality. Thus, both conservation equations (IX2) and (IX7) actually emerge as those expressing the invariance of microscopic theories under specific transformations, involving associated Noether currents. We first discuss the conservation of “particle number” (Sec. IX11)where that denomination has to be taken with a grain of salt, then that of energy and momentum, which in a relativistic context are inseparable (Sec. IX12) IX.11 Particle number conservation The conservation law that was discussed first in the Chapter III introducing the equations of non-relativistic hydrodynamics was that of mass, which, in the case of a single-component fluid, is fully equivalent to the conservation of particle number. In a relativistic system, the number of particles is strictly speaking not conserved, even if the

system is closed. Indeed, thanks to the high kinetic energies available, particle–antiparticle pairs can continuously either be created, or annihilate. If the particles carry some conserved additive quantum numberas e.g electric charge or baryon number, then the difference between the respective amounts of particles and antiparticles is conserved: in a creation resp. annihilation process, both amounts vary simultaneously by +1 resp −1, but the difference remains constant. Accordingly, throughout this Chapter and the following, “particle number” is a shorthand for the difference between the numbers of particles and antiparticles. Similarly, “particle number density” or “particle flux density” also refer to differences between the respective quantities for particles and antiparticles. For the sake of simplicity, we shall consider relativistic fluids comprising a single species of particles, together with their antiparticles, with mass m. IX.11 a Local formulation of

particle number conservation ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: By definition, the local particle (number ) density n(t,~r) in a fluid is such that the product n(t,~r) d3~r represents the number of particles (minus that of antiparticles) in the infinitesimal spatial volume d3~r about position ~r at time t. Since the volume element d3~r depends on the reference frame in which it is measuredremember that in special relativity there is the length contraction phenomenon, this is also the case of the particle density n(t,~r), so that the particle number in d3~r remain independent of the reference frame. Hereafter, n(t,~r) will also be denoted by n(x) The particle flux density ~N (t,~r) is defined in a similar way, as the number of particle that cross a unit surface per unit time interval, where both “unit surface” and “unit time interval” are reference frame-dependent concepts. Together, n(x) and ~N (x) make up a particle number four-current (lxxvii)

N(x), whose Minkowski coordinates at every x are N 0 (x) = c n(x), N i (x) = jNi (x) for i = 1, 2, 3. This is conveniently (lxxvii) (Teilchen-)Viererstrom 135 IX.1 Conservation laws summarized in the form N(x) =   c n(x) ~N (x) or, somewhat improperly,  µ N (x) = (IX.1)  c n(x) . ~N (x) With the help of the particle number four-current, the local formulation of the conservation of particle number in the motion of the system reads, using coordinates dµ N µ (x) = 0, (IX.2a) where dµ ≡ d /dxµ denote the components of the 4-gradient. Denoting the latter, which is a one-form, by d, one may write the even shorter “geometric” (i.e coordinate-invariant) equation d · N(x) = 0, (IX.2b) with d · the four-divergence. Remarks: ∗ Whether N(x) defined by Eq. (IX1) is a 4-vectorthat is, whether it behaves as it should under Lorentz transformationsis at first far from clear. That n(x) d3~r need be a numberie a Lorentz scalar, like d4 x = dx0 d3~rsuggests that

n(x) should transform like the time-like component of a 4-vector. Yet it is admittedly not clear that the associated spatial part should be the particle flux density. We shall see in Sec. IX33 that assuming that there exists a 4-vector field obeying the conservation equation (IX.2) leads in the non-relativistic limit to the above interpretation of its time-like and space-like parts, which may be viewed as a justification.(51) ∗ More generally, one associates to each conserved additive quantum number a 4-current J(x) with components J µ (x), obeying a similar conservation equation d · J(x) = 0 resp. dµ J µ (x) = 0 ∗ If Minkowski coordinates {xµ } are used, the components of the 4-gradient d are simply the partial derivatives ∂µ ≡ ∂ /∂xµ , so that Eq. (IX2a) becomes ∂µ N µ (x) = 0 IX.11 b Global formulation ::::::::::::::::::::::::::: Consider in M 4 a space-like 3-dimensional hypersurface Σi.e a hypersurface at every point of which the normal 4-vector is

time-likewhich extends far enough so that the whole fluid passes through it in its motion; that is, Σ intercepts the worldlines of all fluid particles. 6 6 6 6 6 6 t6 x2 x - 1 Σ Figure IX.1 (51) A better argument is to introduce the particle number 4-current from a microscopic definition, see App. IXA1 136 Fundamental equations of relativistic fluid dynamics The total (net) number N of particles in the fluid is the flux of the particle number 4-current N(x) across Σ Z Z µ 3 (IX.3) N = N (x) d σµ = N(x) · d3 σ, Σ Σ where d3 σµ denotes the components of the 3-hypersurface element d3 σµ ≡ 1p − det g µνρλ dxν dxρ dxλ , 3! (IX.4) with µνρλ the four-dimensional Levi-Civita symbol, with the convention 0123 = +1.(52) Let Ω denote a 4-volume in M 4 , and ∂Ω its 3-surface. Applying the Gauss theorem, the flux of the particle number 4-current across ∂Ω is the integral of the 4-divergence of N(x) over Ω Z I d · N(x) d4 x, (IX.5)

N(x) · d3 σ = ∂Ω Ω where the right member vanishes thanks to the local expression (IX.2) of particle number conservation Splitting ∂Ω into two space-like parts through which particles enter resp leave Ω in their motionthe technical criterion is the sign of N(x) · d3 σ, one finds that there are as many particles that leave as those that enter, which expresses particle number conservation globally. IX.12 Energy-momentum conservation In a relativistic theory, energy and momentum constitute the temporal and spatial components of a four-vector, the four-momentum. To express the local conservationin the absence of external forcesof the latter, the densities and flux densities of energy and momentum at each space-time point x must be combined into a four-tensor
of degree 2, the energy-momentum tensor(lxxviii) also T(x), of type 20 . called stress-energy tensor T This energy-momentum tensor(53) may be defined by the physical content of its 16 Minkowski components T µν (x)

in a given reference frame R: • T 00 (x) is the energy density; • cT 0j (x) is the j-th component of the energy flux density, with j = 1, 2, 3; 1 i0 • T (x) is the density of the i-th component of momentum, with i = 1, 2, 3; c • T ij (x) for i, j = 1, 2, 3 is the momentum flux-density tensor. (IX.6) All physical quantities are to be measured with respect to the reference frame R. Remarks: ∗ The similarity of the notations T resp. T for the energy-momentum four-tensor resp the threedimensional momentum flux-density tensor is not accidental! The former is the natural generalization to the 4-dimensional relativistic framework of the latter, just like four-momentum p, with components pµ is the four-vector associated to the three-dimensional momentum p~. That is, the 3-tensor T is the spatial part of the 4-tensor T , just like the momentum p~ is the spatial part of four-momentum p. ∗ Starting from a microscopic description of the fluid, one can show that the energy-momentum

tensor is symmetric, i.e T µν (x) = T νµ (x) for all µ, ν = 0, 1, 2, 3 (52) This choice is not universal: the alternative convention 0123 = +1 results in 0123 < 0 due to the odd number of minus signs in the signature of the metric tensor. (53) As in the case of the particle number 4-current, the argument showing that T (x) is a Lorentz tensor is to define it microscopically as a tensorsee App. IXA2and to later interpret the physical meaning of the components (lxxviii) Energieimpulstensor 137 IX.2 Four-velocity of a fluid flow Local rest frame In the absence of external force acting on the fluid, the local conservation of the energymomentum tensor reads component-wise dµ T µν (x) = 0 ∀ν = 0, 1, 2, 3, (IX.7a) which represents four equations: the equation with ν = 0 is the conservation of energy, while the equations dµ T µj (x) = 0 for j = 1, 2, 3 are the components of the momentum conservation equation. In geometric formulation, Eq. (IX7a) becomes d · T

(x) = 0. (IX.7b) This is exactly the same form as Eq. (IX2b), just like Eqs (IX2a) and (IX7a) are similar, up to the difference in the tensorial degree of the conserved quantity. As in § IX.11 b, one associates to the energy-momentum tensor T (x) a 4-vector P by Z Z 3 µ P ≡ T (x) · d σ ⇔ P = T µν (x) d3 σν , (IX.8) Σ Σ with Σ a space-like 3-hypersurface. P represents the total 4-momentum crossing Σ, and invoking the Gauss theorem, Eq. (IX7) implies that it is a conserved quantity IX.2 Four-velocity of a fluid flow Local rest frame The four-velocity of a flow is a field, defined at each point x of a space-time domain D, of time-like 4-vectors u(x) with constant magnitude c, i.e [u(x)]2 = uµ (x)uµ (x) = −c2 ∀x, (IX.9) with uµ (x) the (contravariant) components of u(x). At each point x of the fluid, one can define a proper reference frame, the so-called local rest frame,(lxxix) hereafter abbreviated as LR(x), in which the space-like Minkowski components of

the local flow 4-velocity vanish: uµ (x) = (c, 0, 0, 0). (IX.10) LR(x) Let ~v(x) denote the instantaneous velocity of (an observer at rest in) the local rest frame LR(x) with respect to a fixed reference frame R. In the latter, the components of the flow four-velocity ! are γ(x)c , (IX.11) uµ (x) = R γ(x)~v(x) p with γ(x) = 1/ 1 −~v(x)2 /c2 the corresponding Lorentz factor. The local rest frame represents the reference frame in which the local thermodynamic variables of the systemparticle number density n (x) and energy density (x)are defined in their usual sense: n (x) ≡ n(x) , (x) ≡ T 00 (x) . (IX.12) LR(x) LR(x) For the remaining local thermodynamic variables in the local rest frame, it is assumed that they are related to n (x) and (x) in the same way, as when the fluid is at thermodynamic equilibrium. Thus, the pressure P (x) is given by the mechanical equation of state P (x) LR(x) = P ((x), n (x)); (IX.13) the temperature T (x) is given by the thermal

equation of state; the entropy density s(x) is defined by the Gibbs fundamental relation, and so on. (lxxix) lokales Ruhesystem 138 Fundamental equations of relativistic fluid dynamics Remarks: ∗ A slightly more formal approach to define 4-velocity and local rest frame is to turn the reasoning round. Namely, one introduces the latter first as a reference frame LR(x) in which “physics at point x is easy”, that is, in which the fluid is locally motionless. Introducing then an instantaneous inertial reference frame that momentarily coincides with LR(x), one considers an observer O who is at rest in that inertial frame. The four-velocity of the fluid u(x), with respect to some fixed reference frame R, is then the four-velocity of O in Rdefined as the derivative of O’s space-time trajectory with respect to his proper time. The remaining issue is that of the local absence of motion which defines LR(x). In particular, there must be no energy flow, i.e T 0j (x) = 0 One thus

looks for a time-like eigenvector u(x) of the energy-momentum tensor T (x): T (x) · u(x) = −u(x) T µν(x)uν (x) = − uµ (x), ⇔ with − < 0 the corresponding eigenvalue and u(x) normalized to c. Writing that, thanks to the symmetry of T (x), u(x) is also a left-eigenvector, i.e uµ (x)T µν(x) = − uν (x), one finds that the energy flux density vanishes in the reference frame in which the Minkowski components of u(x) have the simple form (IX.10) This constitutes an appropriate choice of local rest frame, and one has at the same time the corresponding four-velocity u(x). ∗ The relativistic energy density  differs from its at first sight obvious non-relativistic counterpart, the internal energy density e. The reason is that  also contains the contribution from the mass energy of the particles and antiparticlesmc2 per (anti)particle, which is conventionally not taken into account in the non-relativistic internal energy density. ∗ To distinguish between the

reference frame dependent quantities, like particle number density n(x) or energy density T 00 (x), and the corresponding quantities measured in the local rest frame, namely n (x) or (x), the latter are referred to as comoving. The comoving quantities can actually be computed easily within any reference frame and coordinate system. Writing thus n (x) ≡ n(x) 1 = N 0 (x) LR(x) c LR(x) = N 0 (x)u0 (x) [u0 (x)]2 = LR(x) N 0 (x)u0 (x) g00 (x)[u0 (x)]2 = LR(x) N µ (x)uµ (x) uν (x)uν (x) , LR(x) where we used that u0 (x) = g00 (x)u0 (x) in the local rest frame, the rightmost term of the above identity is a ratio of two Lorentz-invariant scalars, thus itself a Lorentz scalar field, independent of the reference frame in which it is computed: n (x) = N µ (x)uµ (x) N(x) · u(x) = . uν (x)uν (x) [u(x)]2 (IX.14) Similarly one shows that (x) ≡ T 00 (x) LR(x) = c2 uµ (x)T µν (x)uν (x) [uρ (x)uρ (x)]2 = LR(x) 1 1 T(x)·u(x), (IX.15) uµ (x)T µν (x)uν (x) = 2

u(x)·T 2 c c where the normalization of the 4-velocity was used. In the following Sections, we introduce fluid models, defined by the relations between the conserved currentsparticle number 4-current N(x) and energy-momentum tensor T (x)and the fluid 4-velocity u(x) and comoving thermodynamic quantities. 139 IX.3 Perfect relativistic fluid IX.3 Perfect relativistic fluid By definition, a fluid is perfect when there is no dissipative current in it, see definition (III.16a) As a consequence, one can at each point x of the fluid find a reference frame in which the local properties in the neighborhood of x are spatially isotropic [cf. definition (III23)] This reference frame represents the natural choice for the local rest frame at point x, LR(x). The forms of the particle-number 4-current and the energy-momentum tensor of a perfect fluid are first introduced in Sec. IX31 It is then shown that the postulated absence of dissipative current automatically leads to the conservation of

entropy in the motion (Sec. IX32) Eventually, the low-velocity limit of the dynamical equations is investigated in Sec. IX33 IX.31 Particle four-current and energy-momentum tensor of a perfect fluid To express the defining feature of the local rest frame LR(x), namely the spatial isotropy of the local fluid properties, it is convenient to adopt a Cartesian coordinate system for the space-like directions in LR(x): since the fluid characteristics are the same in all spatial directions, this in particular holds along the three mutually perpendicular axes defining Cartesian coordinates. Adopting momentarily such a systemand accordingly Minkowski coordinates on space-time, the local-rest-frame values of the particle number flux density ~(x), the j-th component cT 0j (x) of the energy flux density, and the density c−1 T i0 (x) of the i-th component of momentum should all vanish. In addition, the momentum flux-density 3-tensor T (x) should also be diagonal in LR(x) All in all, one thus

necessarily has N 0 (x) LR(x) = c n (x), ~(x) LR(x) = ~0, (IX.16a) and T 00 (x) T ij (x) T i0 (x) LR(x) LR(x) LR(x) = (x), = P (x)δ ij , = T 0j (x) LR(x) ∀i, j = 1, 2, 3 = 0, (IX.16b) ∀i, j = 1, 2, 3 where the definitions (IX.12) were taken into account, while P (x) denotes the pressure In matrix form, the energy-momentum tensor (IX.16b) becomes   (x) 0 0 0  0 P (x) 0 0  . (IX.16c) T µν (x) =  0 LR(x) 0 P (x) 0  0 0 0 P (x) Remark: The identification of the diagonal spatial components with a “pressure” term is warranted by the physical interpretation of the T ii (x). Referring to it as “the” pressure anticipates the fact that it behaves as the thermodynamic quantity that is related to energy density and particle number by the mechanical equation of state of the fluid. In an arbitrary reference frame and allowing for the possible use of curvilinear coordinates, the components of the particle number 4-current and the

energy-momentum tensor of a perfect fluid are N µ (x) = n (x)uµ (x) (IX.17a) and   uµ (x) uν (x) T µν (x) = P (x)g µν (x) + (x) + P (x) c2 respectively, with uµ (x) the components of the fluid 4-velocity. (IX.17b) 140 Fundamental equations of relativistic fluid dynamics Relation (IX.17a) resp (IX17b) is an identity between the components of two 4-vectors resp two 4-tensors, which transform identically under Lorentz transformationsi.e changes of reference frameand coordinate basis changes. Since the components of these 4-vectors resp 4-tensors are equal in a given reference framethe local rest frameand a given basisthat of Minkowski coordinates, they remain equal in any coordinate system in any reference frame.  In geometric formulation, the particle number 4-current and energy-momentum tensor respectively read N(x) = n (x)u(x) (IX.18a)   u(x) ⊗ u(x) T (x) = P (x)g−1 (x) + (x) + P (x) . c2 (IX.18b) and The latter is very reminiscent of the 3-dimensional

non-relativistic momentum flux density (III.22); similarly, the reader may also compare the component-wise formulations (III.21b) and (IX17a) Remarks: ∗ The energy-momentum tensor is obviously symmetricwhich is a non-trivial physical statement. For instance, the identity T i0 = T 0i means that (1/c times) the energy flux density in direction i equals (c times) the density of the i-th component of momentumwhere one may rightly argue that the factors of c are historical accidents in the choice of units. This is possible in a relativistic theory only because the energy density also contains the mass energy. ∗ In Eq. (IX17b) or (IX18b), the sum (x) + P (x) is equivalently the enthalpy density w(x)
∗ Equation (IX.17b), (IX18b) or (IX19a) below represents the most general symmetric 20 -tensor that can be constructed using only the metric tensor and the 4-velocity. The component form (IX.17b) of the energy-momentum tensor can trivially be recast as T µν (x) = (x) uµ (x) uν (x)

+ P (x)∆µν (x) c2 (IX.19a) with uµ (x) uν (x) ∆µν (x) ≡ g µν (x) + (IX.19b) c2
the components of a tensor ∆ whichin its 11 -formis actually a projector on the 3-dimensional vector space orthogonal to the 4-velocity u(x), while uµ (x) uν (x)/c2 projects on the time-like direction of the 4-velocity. One easily checks the identities ∆µν (x)∆νρ (x) = ∆µρ (x) and ∆µν (x)uν (x) = 0. From Eq. (IX19a) follows at once that the comoving pressure P (x) can be found in any reference frame as 1 P (x) = ∆µν (x)T µν (x). (IX.20) 3 which complements relations (IX.14) and (IX15) for the particle number density and energy density, respectively. Remark: Contracting the energy-momentum tensor T with the metric tensor twice yields a scalar, the so-called trace of T T (x) : g(x) = T µν (x)gµν (x) = T µµ (x) = 3P (x) − (x). (IX.21) 141 IX.3 Perfect relativistic fluid IX.32 Entropy in a perfect fluid Let s(x) denote the (comoving) entropy density

of the fluid, as defined in the local rest frame LR(x) at point x. IX.32 a Entropy conservation For a perfect fluid, the fundamental equations of motion (IX.2) and (IX7) lead automatically to the local conservation of entropy ::::::::::::::::::::::::::::::   dµ s(x)uµ (x) = 0 (IX.22) with s(x)uµ (x) the entropy four-current. Proof: The relation U = T S − PV + µN N with U resp. µN the internal energy resp the chemical potential gives for the local thermodynamic densities  = T s−P +µNn . Inserting this expression of the energy density in Eq. (IX17b) yields (dropping the x variable for the sake of brevity):   uν uµ uν T µν = P g µν + (T s + µNn ) 2 = P g µν + T (suµ ) + µN (n uµ ) 2 . c c Taking the 4-gradient dµ of this identity gives   dµ uν   ν  uµ uν  µ µ u dµ T µν = dνP + T (suµ )+µN (n uµ ) + + . s d T + n d µ T d (su )+µ d ( n u ) µ µ µ µ N N c2 c2 c2 Invoking the energy-momentum conservation equation (IX.7), the leftmost

member of this identity vanishes The second term between square brackets on the right hand side can be rewritten with the help of the Gibbs–Duhem relation as s dµ T + n dµ µN = dµ P . Eventually, the particle number conservation formulation (IX7) can be used in the rightmost term Multiplying everything by uν yields ν   uν dµuν  uµ uν uν  µ uν u 0 = uν dνP + T (suµ ) + µN (n uµ ) + (d P ) + T d (su ) . µ µ c2 c2 c2 The constant normalization uν uν = −c2 of the 4-velocity implies uν dµuν = 0 for µ = 0, . , 3, so that the equation becomes 0 = uν dνP − (dµ P )uµ − T dµ (suµ ), leading to dµ (suµ ) = 0.  IX.32 b Isentropic distribution ::::::::::::::::::::::::::::::: The local conservation of entropy (IX.22) implies the conservation of the entropy per particle s(x)/n (x) along the motion, where n (x) denotes the comoving particle number density. Proof: the total time derivative of the entropy per particle reads         ∂ s s 1 s

d s ~ = +~v · ∇ = u·d , dt n ∂t n n γ n where the second identity makes use of Eq. (IX11), with γ the Lorentz factor The rightmost     term is then s 1 s 1 s u·d = u · ds − 2 u · dn = u · ds − u · dn . n n n n n The continuity equation d · (n u) = 0 gives u · dn = −n d · u, implying    
d s 1 1 1 s = u·d = u · ds + s d · u = d · (su) = 0, dt n γ n γn γn where the last identity expresses the conservation of entropy.  142 Fundamental equations of relativistic fluid dynamics IX.33 Non-relativistic limit We shall now consider the low-velocity limit |~v|  c of the relativistic equations of motion (IX.2) and (IX.7), in the case when the conserved currents are those of perfect fluids, namely as given by relations (IX.17a) and (IX17b) Anticipating on the result, we shall recover the equations governing the dynamics of non-relativistic perfect fluids presented in Chapter III, as could be expected for the sake of consistency. In the

small-velocity limit, the typical velocity of the atoms forming the fluid is also much smaller than the speed of light, which has two consequences. On the one hand, the available energies are too low to allow the creation of particle–antiparticle pairswhile their annihilation remains possible, so that the fluid consists of either particles or antiparticles. Accordingly, the “net” particle number density n (x), difference of the amounts of particles and antiparticles in a unit volume, actually coincides with the “true” particle number density. On the other hand, the relativistic energy density  can then be expressed as the sum of the contribution from the (rest) masses of the particles and of a kinetic energy term. By definition, the latter is the local internal energy density e of the fluid, while the former is simply the number density of particles multiplied by their mass energy: (x) = n (x)mc2 + e(x) = ρ(x)c2 + e(x), (IX.23) with ρ(x) the mass density of the fluid

constituents. It is important to note that the internal energy density e is of order ~v2 /c2 with respect to the mass-energy term. The same holds for the pressure P , which is of the same order of magnitude as e.(54) Eventually, Taylor expanding the Lorentz factor associated with the flow velocity yields   ~v(x)4 1 ~v(x)2 γ(x) ∼ 1 + . (IX.24) +O 2 c2 c4 |~v|c Accordingly, to leading order in ~v2 /c2 , the components (IX.11) of the flow 4-velocity read ! c µ . (IX.25) u (x) ∼ |~v|c ~v(x) Throughout the Section, we shall omit for the sake of brevity the variables x resp. (t, ~r) of the various fields. In addition, we adopt for simplicity a system of Minkowski coordinates IX.33 a Particle number conservation :::::::::::::::::::::::::::::::::::::: The 4-velocity components (IX.25) give for those of the particle number 4-current (IX17a) ! nc µ . N ∼ |~v|c n~v Accordingly, the particle number conservation equation (IX.2) becomes 3 1 ∂(n c) X ∂(n vi ) ∂n ~ 0 = ∂µ N

≈ + + ∇ · (n ~v). = i c ∂t ∂x ∂t µ (IX.26) i=1 That is, one recovers the non-relativistic continuity equation (III.10) IX.33 b Momentum and energy conservation :::::::::::::::::::::::::::::::::::::::::::::: The (components of the) energy-momentum tensor of a perfect fluid are given by Eq. (IX17b) Performing a Taylor expansion including the leading and next-to-leading terms in |~v|/c yields, under consideration of relation (IX.23) (54) This is exemplified for instance by the non-relativistic classical ideal gas, in which the internal energy density is e = nc V kB T with c V a number of order 1this results e.g from the equipartition theoremwhile its pressure is P = n kB T . 143 IX.3 Perfect relativistic fluid T 00 T 0j T ij  2 ~v = −P + γ (ρc + e + P ) ∼ ρc + e + ρ~v + O 2 ; c |~v|c 2 2 2 2 (IX.27a)  3
j vj |~v| j 2 v = T = γ (ρc + e + P ) ∼ ρcv + e + P + ρ~v +O 3 ; c |~v|c c c   2  ~v ~v2 vi vj ij 2 2 ij ij i j = P g + γ (ρc +

e + P ) 2 ∼ P g + ρ v v + O 2 = T + O 2 . c |~v|c c c j0 2 2 (IX.27b) (IX.27c) In the last line, we have introduced the components Tij , defined in Eq. (III21b), of the threedimensional momentum flux-density tensor for a perfect non-relativistic fluid As emphasized below Eq. (IX23), the internal energy density and pressure in the rightmost terms of the first or second equations are of the same order of magnitude as the term ρ~v2 with which they appear, i.e they are always part of the highest-order term. Momentum conservation Considering first the components (IX.27b), (IX27c), the low-velocity limit of the relativistic momentum-conservation equation ∂µ T µj = 0 for j = 1, 2, 3 reads  2  2 3 3 Tij Tij ~v ~v 1 ∂(ρcvj ) X ∂T ∂(ρvj ) X ∂T 0= (IX.28) + + +O 2 = +O 2 . i i c ∂t ∂x c ∂t ∂x c i=1 i=1 This is precisely the conservation-equation formulation (III.24a) of the Euler equation in absence of external volume forces. Energy conservation Given the

physical interpretation of the components T 00 , T i0 with i = 1, 2, 3, the component ν = 0 of the energy-momentum conservation equation (IX.7), ∂µ T µ0 = 0, should represent the conservation of energy. As was mentioned several times, the relativistic energy density and flux density actually also contain a term from the rest mass of the fluid constituents. Thus, the leading order contribution to ∂µ T µ0 = 0, coming from the first terms in the right members of Eqs. (IX27a) and (IX27b), is  2 3 ~v ∂(ρc) X ∂(ρcvi ) + +O 2 , 0= i ∂t ∂x c i=1 that is, up to a factor c, exactly the continuity equation (III.9), which was already shown to be the low-velocity limit of the conservation of the particle-number 4-current. To isolate the internal energy contribution, it is thus necessary to subtract that of mass energy. In the fluid local rest frame, relation (IX.23) shows that one must subtract ρc2 from  The former simply equals ρcu0 |LR , while the latter is the component

µ = 0 of T µ0 |LR , whose space-like components vanish in the local rest frame. To fully subtract the mass energy contribution in any frame from both the energy density and flux density, one should thus consider the 4-vector T µ0 − ρcuµ . Accordingly, instead of simply using ∂µ T µ0 = 0, one should start from the equivalentthanks to Eq. (IX2) and the relation ρ = mn equation ∂µ (T µ0 − ρcuµ ) = 0 With the approximations  2 ~v 1 2 0 2 2 ρcu = γρc = ρc + ρ~v + O 2 2 c and j j j ρcu = γρcv = ρcv +    5 |~v| 1 2 vj ρ~v +O 3 2 c c one finds 0 = ∂µ T µ0 µ − ρcu
  X   j  2 3 ~v v ∂ 1 2 1∂ 1 2 ρ~v + e + ρ~v + e + P +O 2 , = j c ∂t 2 ∂x 2 c c j=1 144 that is Fundamental equations of relativistic fluid dynamics      ∂ 1 2 1 2 ~ ρ~v + e + ∇ · ρ~v + e + P ~v ≈ 0. ∂t 2 2 (IX.29) This is the non-relativistic local formulation of energy conservation (III.33) for a perfect fluid in absence of external

volume forces. Since that equation had been postulated in Section III41, the above derivation may be seen as its belated proof. IX.33 c Entropy conservation :::::::::::::::::::::::::::::: Using the approximate 4-velocity components (IX.25), the entropy conservation equation (IX22) becomes in the low-velocity limit 3 1 ∂(sc) X ∂(svi ) ∂s ~ 0 = ∂µ (su ) ≈ + + ∇ · (s~v), = i c ∂t ∂x ∂t µ (IX.30) i=1 i.e gives the non-relativistic equation (III34) IX.4 Dissipative relativistic fluids In a dissipative relativistic fluid, the transport of particle number and 4-momentum is no longer only convectivei.e caused by the fluid motion, but may also diffusive, due eg to spatial gradients of the flow velocity field, the temperature, or the chemical potential(s) associated with the conserved particle number(s). The description of these new types of transport necessitate the introduction of additional contributions to the particle-number 4-current and the energy-momentum tensor

(Sec. IX41), that break the local spatial isotropy of the fluid As a matter of fact, the local rest frame of the fluid is no longer uniquely, but there are in general different choices that lead to “simple” expressions for the dynamical quantities (Sec. IX42) For the sake of brevity, we adopt in this Section a “natural” system of units in which the speed of light c and the Boltzmann constant kB equal 1. IX.41 Dissipative currents To account for the additional types of transport present in dissipative fluids, extra terms are added to the particle-number 4-current and energy-momentum tensor. Denoting with a subscript (0) the quantities for a perfect fluid, their equivalent in the dissipative case thus read µ N µ (x) = N(0) (x) + nµ (x) , µν T µν(x) = T(0) (x) + τ µν(x) (IX.31a) T (x) = T(0)(x) + τ (x) (IX.31b) or equivalently, in geometric formulation N(x) = N(0)(x) + n(x) , with n(x) resp. τ (x) a 4-vector resp 4-tensor of degree 2, with components nµ (x)

resp τ µν (x), that represents a dissipative particle-number resp. energy-momentum flux density In analogy by the perfect-fluid case, it is natural to introduce a 4-velocity u(x) in terms of which the quantities n(0)(x), T(0)(x) have a simple, “isotropic” expression. Accordingly, let u(x) be an arbitrary time-like 4-vector field with constant magnitude −c2 = −1, with components uµ (x), µ ∈ {0, 1, 2, 3}. The reference frame in which the spatial components of this “4-velocity” vanishes will constitute the local rest frame LR(x) associated with u(x). The projector ∆ on the 3-dimensional vector space orthogonal to the 4-velocity u(x) is defined as in Eq. (IX19b), ie has components ∆µν(x) ≡ g µν(x) + uµ (x)uν(x), (IX.32) with g µν(x) the components of the inverse metric tensor g−1 (x). For the comprehension it is important to realize that ∆ plays the role of the identity in the 3-space orthogonal to u(x) 145 IX.4 Dissipative relativistic fluids

In analogy with Eqs. (IX17a), (IX18), and (IX19a), one thus writes N µ (x) = n (x)uµ (x) + nµ (x) (IX.33a) N(x) = n (x)u(x) + n(x) (IX.33b) T µν(x) = (x)uµ (x)uν (x) + P (x)∆µν(x) + τ µν(x) (IX.34a) T(x) = (x)u(x)⊗ u(x) + P (x)∆ ∆(x) + τ(x). (IX.34b) or equivalently and i.e, in geometric form The precise physical content and mathematical form of the additional terms can now be further specified. Tensor algebra ::::::::::::::: In order for n (x) to represent the (net) comoving particle density, the dissipative 4-vector n(x) may have no timelike component in the the local rest frame LR(x) defined by the 4-velocity, see definition (IX.12) Accordingly, the condition uµ (x)nµ (x) LR(x) =0 must hold in the local rest frame. Since the left hand side of this identity is a Lorentz scalar, it holds in any reference frame or coordinate system: uµ (x)nµ (x) = u(x) · n(x) = 0. (IX.35a) Equations (IX.33a), (IX33) thus represent the decomposition of a

4-vector in a component parallel to the flow 4-velocity and a component orthogonal to it. In keeping, one can write nµ (x) = ∆µν(x)Nν (x). (IX.35b) Physically, n(x) represents a diffusive particle-number 4-current in the local rest frame, which describes the non-convective transport of particle number. Similarly, the dissipative energy-momentum current $(x) can have no 00-component in the local rest frame, to ensure that T 00 (x) in that frame still define the comoving energy density (x). This means that the components τ µν (x) may not be proportional to the product uµ (x)uν (x). The most general symmetric tensor of degree 2 which obeys that condition is of the form τ µν(x) = q µ (x)uν(x) + q ν(x)uµ (x) + π µν(x), (IX.36a) with q µ (x) resp. π µν(x) the components of a 4-vector q(x) resp π (x) such that uµ (x)q µ (x) = u(x) · q(x) = 0 (IX.36b) uµ (x)π µν (x)uν (x) = u(x) · π (x) · u(x) = 0. (IX.36c) and Condition (IX.36b) expresses that

q(x) is a 4-vector orthogonal to the 4-velocity u(x), which physically represents the heat current or energy flux density in the local rest frame In turn, the symmetric tensor π (x) can be decomposed into the sum of a traceless tensor $ (x) with components $µν(x) and a tensor proportional to the projector (IX.19b) orthogonal to the 4-velocity π µν(x) = $µν(x) + Π(x)∆µν(x). (IX.36d) 146 Fundamental equations of relativistic fluid dynamics The tensor $ (x) is the shear stress tensor in the local rest frame of the fluid, that describes the transport of momentum due to shear deformations. Eventually, Π(x) represents a dissipative pressure term, since it behaves as the thermodynamic pressure P (x) as shown by Eq. (IX37) below All in all, the components of the energy-momentum tensor in a dissipative relativistic fluid may thus be written as   T µν(x) = (x)uµ (x)uν (x) + P (x) + Π(x) ∆µν(x) + q µ (x)uν(x) + q ν(x)uµ (x) + $µν(x), (IX.37a) which in

geometric formulation reads   T (x) = (x)u(x)⊗ u(x) + P (x) + Π(x) ∆ (x) + q(x)⊗ u(x) + u(x)⊗ q(x) + $ (x). (IX.37b) One can easily check the identities q µ (x) = ∆µν (x)Tνρ (x)uρ (x);   2 1 µ ∆ ρ (x)∆νσ (x) + ∆νρ (x)∆µσ (x) − ∆µν (x)∆ρσ (x) T ρσ (x); $µν (x) = 2 3 1 P (x) + Π(x) = − ∆µν (x)Tµν (x), 3 (IX.38a) (IX.38b) (IX.38c) which together with Eq. (IX15) (x) = uµ (x)T µν (x)uν (x) = u(x) · T (x) · u(x) (IX.38d) allow one to recover the various fields in which the energy-momentum tensor has been decomposed. Remarks: ∗ The energy-momentum tensor comprises 10 unknown independent fields, namely the components T µν with ν ≥ µ. In the decomposition (IX37), written in the local rest frame, (x), P (x) + Π(x), the space-like components q i (x) and $ij (x) represent 1+1+3+5=10 equivalent independent fieldsout of the 6 components $ij (x) with j ≥ i, one of the diagonal ones is fixed by the condition on the

trace. This in particular shows that the decomposition of the left hand side of Eq (IX38c) into two terms is as yet prematurethe splitting actually requires of an equation of state to properly identify P (x). Similarly, the 4 unknown components N µ of the particle-number 4-current are expressed in terms of n (x) and the three spatial components ni (x), i.e an equivalent number of independent fields
∗ Let aµν denote the (contravariant) components of an arbitrary 20 -tensor. One encounters in the literature the various notations
1 a(µν) ≡ aµν + aνµ , 2 which represents the symmetric part of the tensor,
1 a[µν] ≡ aµν − aνµ 2 for the antisymmetric partso that aµν = a(µν) + a[µν] , and   1 µν hµνi (µ ν) a ≡ ∆ρ ∆ σ − ∆ ∆ρσ aρσ , 3 which is the symmetrized traceless projection on the 3-space orthogonal to the 4-velocity. Using these notations, the dissipative stress tensor (IX.36a) reads τ µν(x) = q (µ (x)uν)(x) + $µν(x) −

Π(x)∆µν(x), while Eq. (IX38b) becomes $µν(x) = T hµνi(x) 147 IX.4 Dissipative relativistic fluids IX.42 Local rest frames At a given point in a dissipative relativistic fluid, the net particle number(s) and the energy can flow in different directions. This can happen in particular because particle–antiparticle pairs, which do not contribute to the net particle-number density, still transport energy. Another, not exclusive, possibility is that different conserved quantum numbers flow in different directions. In any case, one can in general not find a preferred reference frame in which the local properties of the fluid are isotropic. As a consequence, there is also no unique “natural” choice for the 4-velocity u(x) of the fluid motion. On the contrary, several definitions of the flow 4-velocity are possible, which imply varying relations for the dissipative currents, although the physics that is being described remains the same. • A first natural possibility,

proposed by Eckart(at) [44], is to take the 4-velocity proportional to the particle-number 4-current,(55) namely N µ (x) uµEckart(x) ≡ p Nν (x)N ν (x) . (IX.39) Accordingly, the dissipative particle-number flux n(x) vanishes automatically, so that the expression of particle-number conservation is simpler with that choice. The local rest frame associated with the flow 4-velocity (IX.39) is then referred to as Eckart frame A drawback of that definition of the fluid 4-velocity is that the net particle number can possibly vanish in some regions of a given flow, so that uEckart (x) is not defined unambiguously in such domains. • An alternative natural definition is that of Landau(au) (and Lifshitz(av) ), according to whom the fluid 4-velocity is taken to be proportional to the energy flux density. The corresponding 4-velocity is defined by the implicit equation T µν (x)uνLandau(x) uµLandau(x) = q uλLandau(x)Tλρ(x)Tρσ (x)uσLandau(x) (IX.40a) or equivalently

uµLandau(x) = q T µν (x)uνLandau(x) uρLandau(x)Tρσ (x)uσLandau(x) . (IX.40b) With this choice, which in turn determines the Landau frame, the heat current q(x) vanishes, so that the dissipative tensor τ (x) satisfies the condition uµLandau(x)τµν (x) = 0 (IX.40c) and reduces to its “viscous” part π (x). For a fluid without conserved quantum number, the Landau definition of the 4-velocity is the only natural one. However, in the presence of a conserved quantum number, heat conduction now enters the dissipative part of the associated current n(x), which conflicts with the intuition gained in the non-relativistic case. This implies that the Landau choice does not lead to a simple behavior in the limit of low velocities. (55) (at) . or to one of the quantum-number 4-currents, in case there are several conserved quantum numbers C. Eckart, 1902–1973 1915–1985 (au) L. D Landau = L. D Landau, 1908–1968 (av) E. M Lifxic = E. M Lifshitz, 148

Fundamental equations of relativistic fluid dynamics Eventually, one may of course choose to work with a general 4-velocity u(x), and thus to keep both the diffusive particle-number current and the heat flux density in the dynamical fields (IX.33)– (IX.37) IX.43 General equations of motion By substituting the decompositions (IX.33), (IX37) into the generic conservation laws (IX2), (IX.7), one can obtain model-independent equations of motion, that do not depend on any assumption on the various dissipative currents For that purpose, let us introduce the notation ∇µ (x) ≡ ∆µν(x)dν , (IX.41a) where dν , ν ∈ {0, 1, 2, 3} denotes the components of the 4-gradient dinvolving covariant derivatives in case a non-Minkowski system of coordinates is being used. In geometric formulation, this definition reads ∇ (x) ≡ ∆ (x) · d. (IX.41b) As is most obvious in the local rest frame at point x, in which the timelike component ∇0 (x) vanishes, ∇ (x) is the projection of the

gradient on the space-like 3-space orthogonal to the 4-velocity. Let us further adopt the Landau definition for the flow 4-velocity,(56) which is simply denoted by u(x) without subscript. The net particle-number conservation equation (IX.2) first yields dµ N µ (x) = uµ (x)dµ n (x) + n (x)dµ uµ (x) + dµ nµ (x) = 0. (IX.42a) In turn, the conservation of the energy momentum tensor (IX.7), projected perpendicular to resp along the 4-velocity, gives   (IX.42b) ∆ρν (x)dµ T µν (x) = (x) + P (x) uµ (x)dµ uρ (x) + ∇ρ (x)P (x) + ∆ρν (x)dµ π µν (x) = 0 resp.   uν (x)dµ T µν (x) = −uµ (x)dµ (x) − (x) + P (x) dµ uµ (x) + uν (x)dµ π µν (x) = 0. In the latter equation, one can substitute the rightmost term by       uν (x)dµ π µν (x) = dµ uν (x)π µν (x) − dµ uν (x) π µν (x) = − dµ uν (x) π µν (x), where the second equality follows from condition (IX.40c) with τ µν = π µν (since q = 0) Using the identity dµ =

uµ (u · d) + ∇µ , and again the condition uν π µν = 0, this becomes   uµ (x)dµ (x) + (x) + P (x) dµ uµ (x) + π µν (x)∇µ (x)uν (x) = 0. (IX.42c) Equations (IX.42a)–(IX42c) represent the relations governing the dynamics of a dissipative fluid in the Landau frame. Remark: If one adopts Eckart’s choice of velocity, the resulting equations of motion differ from those given herefor instance, the third term d · n(x) in Eq. (IX42a) drops out, since n(x) = 0, yet they are physically totally equivalent. Entropy law in a dissipative relativistic fluid ::::::::::::::::::::::::::::::::::::::::::::: Combining the dynamical equation (IX.42c) with the thermodynamic relations + P = T s+µNn and d = T ds + µN dn , one finds   T (x)dµ s(x)uµ (x) = −π µν (x)∇µ (x)uν (x) + µN (x)dµ nµ (x) (56) This choice of form for u(x) is often announced as “let us work in the Landau frame”, where frame is to be understood in its sense of framework. IX.4

Dissipative relativistic fluids or equivalently, using the identity nµ dµ = nµ ∇µ that follows from nµ uµ = 0,     ∇µ (x)uν (x) µ (x) µ (x) . dµ s(x)uµ (x) − N nµ (x) = −π µν (x) − nµ (x)∇µ N T (x) T (x) T (x) 149 (IX.43a) Using the symmetry of π µν , one can replace ∇µ uν by its symmetric part 21 (∇µ uν + ∇ν uµ ) in the first term on the right hand side. With the decompositions π µν = $µν + Π∆µν [Eq (IX36d)] and  



1 1 1 2 1 ∇µ uν + ∇ν uµ = ∇µ uν + ∇ν uµ − ∆µν ∇ · u + ∆µν ∇ · u ≡ S µν + ∆µν ∇ · u , 2 2 3 3 3 where the S µν are the components of a traceless tensor(57) comparing with Eq. (II15d), this is the rate-of-shear tensor, while ∇ · u is the (spatial) 3-divergence of the 4-velocity field, one finds     µN (x) µ $µν (x) Π(x) µN (x) µ µ . (IX43b) dµ s(x)u (x) − n (x) = − S µν (x) − ∇ (x) · u(x) − n (x)∇µ T (x) T (x) T (x) T (x) The

left member of this equation is the 4-divergence of the entropy 4-current S(x), with components S µ (x), comprising on the one hand the convective transport of entropywhich is the only contribution present in the perfect-fluid case, see Eq. (IX22), and on the other hand a contribution from the dissipative particle-number current. Remark: When working in the Eckart frame, the dissipative particle-number current no longer contributes to the entropy 4-current Swhich is obvious since n vanishes in that frame!, but the heat 4-current q does. In an arbitrary frameie using a different choice of fluid 4-velocity and thereby of local rest frame, both n and q contribute to S and to the right hand side of Eq. (IX43b) Let Ω be the 4-volume that represents the space-time trajectory of the fluid between an initial and a final times. Integrating Eq (IX43b) over Ω while using the same reasoning as in § IX11 b, one sees that the left member will yield the change in the total entropy of the fluid

during these two times. This entropy variation must be positive to ensure that the second law of thermodynamics holds. Accordingly, one requests that the integrand be positive: dµ S µ (x) ≥ 0 This requirement can be used to build models for the dissipative currents. IX.44 First order dissipative relativistic fluid dynamics The decompositions (IX.33), (IX37) are purely algebraic and do not imply anything regarding the physics of the fluid. Any such assumption involve two distinct elements: an equation of state, relating the energy density  to the (thermodynamic) pressure P and the particle-number density n ; and a constitutive equation (lxxx) that models the dissipative effects, i.e the diffusive particle-number 4-current N(x), the heat flux density q(x) and the dissipative stress tensor τ(x). Several approaches are possible to construct such constitutive equations. A first one would be to compute the particle-number 4-current and energy-momentum tensor starting from an

underlying microscopic theory, in particular from a kinetic description of the fluid constituents. Alternatively, one can work at the “macroscopic” level, using the various constraints applying to such A first constraint is that the tensorial structure of the various currents should be the correct one: using as building blocks the 4-velocity u, the 4-gradients of the temperature T , the chemical potential µ, and of u, as well as the projector ∆ , one writes the possible forms of n, q, Π, and $ . A further condition is that the second law of thermodynamics should hold, i.e that when inserting the dissipative currents in Eq. (IX43b), one obtains a 4-divergence of the entropy 4-current that is always positive. (57) In the notation introduced in the remark at the end of Sec. IX41, S µν = ∇hµ uνi (lxxx) konstitutive Gleichung 150 Fundamental equations of relativistic fluid dynamics Working like in Sec. IX43 in the Landau frame,(58) in which the heat flux density q(x)

vanishes, the simplest possibility that satisfies all constraints is to require Π(x) = −ζ(x)∇µ (x)uµ (x) (IX.44a) for the dissipative pressure,    2 µν  ρ µ ν ν µ Sµν(x) $ (x) = −η(x) ∇ (x)u (x) + ∇ (x)u (x) − ∆ (x) ∇ (x)uρ (x) = −2η(x)S 3 µν (IX.44b) for the shear stress tensor, and     n (x)T (x) 2 µ µN (x) n (x) = κ(x) ∇ (x) (x)+ P (x) T (x) µ (IX.44c) for the dissipative particle-number 4-current, with η, ζ, κ three positive numberswhich depend on the space-time position implicitly, inasmuch as they vary with temperature and chemical potential. The first two ones are obviously the shear and bulk viscosity coefficients, respectively, as hinted at by the similarity with the form (III.26f) of the shear stress tensor of a Newtonian fluid in the non-relativistic case. Accordingly, the equation of motion (IX42b) in which the dissipative stress tensor is substituted by π µν = $µν + Π∆µν with the forms (IX.44a), (IX44b)

yields the relativistic version of the Navier–Stokes equation. What is less obvious is that κ in Eq. (IX44c) does correspond to the heat conductivitywhich explains why the coefficient in front of the gradient is written in a rather contrived way. Inserting the dissipative currents (IX.44) in the entropy law (IX43b), the latter becomes   $(x) $ (x) :$ Π(x)2 (x)+ P (x) 2 n(x)2 d · S(x) = + + . (IX.45) 2η(x)T (x) ζ(x)T (x) n (x)T (x) κ(x)T (x) Since n(x) is space-like, the right hand side of this equation is positive, as it should. The constitutive equations (IX.44) only involve first order terms in the derivatives of velocity, temperature, or chemical potential. In keeping, the theory constructed with such Ansätze is referred to as first order dissipative fluid dynamicswhich is the relativistic generalization of the set of laws valid for Newtonian fluids. This simple relation to the non-relativistic case, together with the fact that only 3 transport coefficients are

neededwhen working in the Landau or Eckart frames, in the more general case, one needs 4 coefficientsmakes first-order dissipative relativistic fluid dynamics attractive. The theory suffers however from a severe issue, which does not affect its non-relativistic counterpart. Indeed, it has been shown that many solutions of the relativistic Navier–Stokes(–Fourier) equations are unstable against small perturbations [46]. Such disturbances will grow exponentially with time, on a microscopic typical time scale. As a result, the velocity of given modes can quickly exceed the speed of light, which is of course unacceptable in a relativistic theory. In addition, gradients also grow quickly, leading to the breakdown of the small-gradient assumption that underlies the construction of first-order dissipative fluid dynamics. This exponential growth of perturbation is especially a problem for numerical implementations of the theory, in which rounding errors which quickly propagate. Violations

of causality actually occur for short-wavelength modes, which from a physical point of view should not be described by fluid dynamics since they involve length scales on which the system is not “continuous”. As such, the issue is more mathematical than physical These modes (58) The corresponding formulae for Π, $µν and q µ valid in the Eckart frame, in which n vanishes, can be found e.g in Ref. [45, Sec 24] 151 IX.4 Dissipative relativistic fluids do however play a role in numerical computations, so that there is indeed a problem when one is not working with an analytical solution. As a consequence, including dissipation in relativistic fluid dynamics necessitates going beyond a first-order expansion in gradients, i.e beyond the relativistic Navier–Stokes–Fourier theory IX.45 Second order dissipative relativistic fluid dynamics Coming back to an arbitrary 4-velocity u(x), the components of the entropy 4-current S(x) in a first-order dissipative theory read S µ (x)

= P (x)g µν(x) − T µν(x) T (x) uν (x) − µN (x) µ N (x), T (x) (IX.46a) or equivalently S µ (x) = s(x)uµ (x) − µN (x) µ 1 µ n (x) + q (x) T (x) T (x) (IX.46b) which simplify to the expression between square brackets on the left hand side of Eq. (IX43b) with Landau’s choice of 4-velocity. This entropy 4-current is linear in the dissipative 4-currents n(x) and q(x). In addition, it is independent of the velocity 3-gradientsencoded in the expansion rate ∇ (x)·u(x) and the rate-ofshear tensor S (x), which play a decisive role in dissipation. That is, the form (IX46) can be generalized. A more general form for the entropy 4-current is thus 1 1 µ (x) q(x) + Q(x) (IX.47a) S(x) = s(x)u(x) − N n(x) + T (x) T (x) T (x) or equivalently, component-wise, S µ (x) = s(x)uµ (x) − µN (x) µ 1 µ 1 n (x) + q (x) + Qµ (x), T (x) T (x) T (x) (IX.47b) with Q(x) a 4-vector, with components Qµ (x), that depends on the flow 4-velocity and its gradients where ∇ (x)

· u(x) and S (x) are traditionally replaced by Π(x) and $ (x)and on the dissipative currents:
Qµ (x) = Qµ u(x), n(x), q(x), Π(x), $ (x) . (IX.47c) In second order dissipative relativistic fluid dynamics, the most general form for the additional 4-vector Q(x) contributing to the entropy density is [47, 48, 49] Q(x) = $(x) :$ $(x) β0 (x)Π(x)2 + β1 (x)qN (x)2 + β2 (x)$ α0 (x) α1 (x) u(x) − Π(x)qN (x) − $ (x)·qN (x), 2T (x) T (x) T (x) (IX.48a) where qN (x) ≡ q(x) − (x) + P (x) n(x); n (x) component-wise, this reads β0 (x)Π(x)2 + β1 (x)qN (x)2 + β2 (x)$νρ (x)$νρ(x) µ α1 (x) µ α0 (x) u (x) − Π(x)qNµ (x)− $ (x)qNρ (x). 2T (x) T (x) T (x) ρ (IX.48b) The 4-vector Q(x) is now quadratic (“of second order”) in the dissipative currentsin the wider senseq(x), n(x), Π(x) and $ (x), and involves 5 additional coefficients depending on temperature and particle-number density, α0 , α1 , β0 , β1 , and β2 . Substituting this form of Q(x) in the

entropy 4-current (IX.47), the simplest way to ensure that its 4-divergence be positive is to postulate linear relationships between the dissipative currents Qµ (x) = 152 Fundamental equations of relativistic fluid dynamics and the gradients of velocity, chemical potential (or rather of −µN /T ), and temperature (or rather, 1/T ), as was done in Eqs. (IX44) This recipe yields differential equations for Π(x), $ (x), qN (x), representing 9 coupled scalar equations of motion. These describe the relaxationwith appropriate characteristic time scales τΠ , τ$ , τqN respectively proportional to β0 , β2 , β1 , while the involved “time derivative” is that in the local rest frame, u · d, of the dissipative currents towards their first-order expressions (IX.44) Adding up the new equations to the usual ones (IX.2) and (IX7), the resulting set of equations, known as (Müller(aw) –)Israel(ax) –Stewart(ay) theory, is no longer plagued by the issues that affects the

relativistic Navier–Stokes–Fourier equations. Bibliography for Chapter IX • Andersson & Comer [50]; • Landau–Lifshitz [3, 4], Chapter XV, § 133,134 (perfect fluid) and § 136 (dissipative fluid); • Romatschke [51]; • Weinberg [52], Chapter 2, § 10 (perfect fluid) and § 11 (dissipative fluid). (aw) I. Müller, born 1936 (ax) W. Israel, born 1931 (ay) J. M Stewart, born 1943 Appendices to Chapter IX IX.A Microscopic formulation of the hydrodynamical fields In Sec. IX1, we have taken common non-relativistic quantitiesparticle number density and flux density, energy density, momentum flux density, and so onand claimed that they may be used to define a 4-vector resp. a Lorentz tensor, namely the particle number 4-current N(x) resp the energymomentum tensor T (x) However, we did not explicitly show that the latter are indeed a 4-vector resp. a tensor For that purpose, the best is to turn the reasoning round and to introduce quantities which are manifestly, by

construction, a Lorentz 4-vector or tensor. In turn, one investigates the physical interpretation of their components and shows that it coincides with known non-relativistic quantities. Throughout this Appendix, we consider a system Σ of N “particles”i.e carriers of some conserved additive quantum numberlabeled by k ∈ {1, , N } with world-lines xk (τ ) and associated 4-velocities uk (τ ) ≡ dxk (τ )/dτ , where the scalar parameter τ along the world-line of a given particle is conveniently taken as its proper time. IX.A1 Particle number 4-current The particle-number 4-current associated with the collection of particles Σ is defined as N Z X
uk (τ )δ (4) x−xk (τ ) d(cτ ) (IX.A1a) N(x) ≡ k=1 or component-wise N µ (x) ≡ N Z X
uµk (τ )δ (4) xν−xνk (τ ) d(cτ ) for µ = 0, 1, 2, 3, (IX.A1b) k=1 where the k-th integral in either sum is along the world-line of particle k. The right hand sides of these equations clearly define a 4-vector resp. its

components For the latter, some simple algebra yields the identities N X
1 0 N (t,~r) = δ (3) ~r − ~xk (t) , (IX.A2a) c k=1 N i (t,~r) = N X vki (t)δ (3) ~r − ~xk (t)
(IX.A2b) k=1 with ~xk (t) the spatial trajectory corresponding to the world-line xk (τ ). Using u0k (τ ) = c dtk (τ )/dτ and changing the parameter along the world-lines from τ to t, one finds N Z N Z X X
(3)
dtk (τ )

0 N (t,~r) = c δ ct−ctk (τ ) δ ~x −~xk (τ ) d(cτ ) = c δ t−tk (t) δ (3) ~x −~xk (t) dt dτ k=1 i.e N 0 (t,~r) = c k=1 N X δ (3) ~x −~xk (t) . The proof for Eq (IXA2b) is identical
 k=1 Inspecting the right hand sides of relations (IX.A2), they obviously represent the particle number density and flux density for the system Σ, respectively 154 Fundamental equations of relativistic fluid dynamics IX.A2 Energy-momentum tensor Denoting by pk the 4-momentum carried by particle k, the energy-momentum tensor associated with the collection of particles

Σ is defined as N Z X
pk (τ ) ⊗ uk (τ )δ (4) x−xk (τ ) d(cτ ) (IX.A3a) T (x) ≡ k=1 where the k-th integral in the sum is along the world-line of particle k, as above; component-wise, this gives N Z X
µν T (x) ≡ pµk (τ )uνk (τ )δ (4) xλ−xλ (τ ) d(cτ ) for µ, ν = 0, 1, 2, 3. (IX.A3b) k=1 The members of these equations clearly define a Lorentz tensor of type 2 0
resp. its components Repeating the same derivation as that leading to Eq. (IXA2a), one shows that T µ0 (t,~r) = N X
pµk (t)cδ (3) ~r − ~xk (t) . (IX.A4a) k=1 Recognizing in p0k c the energy of particle k, T 00 represents the energy density of the system Σ under the assumption that the potential energy associated with the interaction between particles is much smaller than their mass and kinetic energies, while T i0 for i = 1, 2, 3 represents c times the density of the i-th component of momentum. In turn, 0j T (t,~r) = N X p0k (t)vkj (t)δ (3) ~r − ~xk (t)
(IX.A4b) k=1

with j ∈ {1, 2, 3} is the 1/c times the j-th of the energy flux density of the collection of particles. Eventually, for i, j = 1, 2, 3 T ij (t,~r) = N X
pik (t)vkj (t)δ (3) ~r − ~xk (t) (IX.A4c) k=1 is clearly the j-th component of the flux density of momentum along the i-th direction. Remark: Invoking the relation p = mu between the 4-momentum, mass and 4-velocity of a (massive!) particle shows at once that the energy-momentum tensor (IX.A3) is symmetric IX.B Relativistic kinematics Later IX.C Equations of state for relativistic fluids C HAPTER X Flows of relativistic fluids X.1 Relativistic fluids at rest X.2 One-dimensional relativistic flows X.21 Landau flow [53, 54] X.22 Bjorken flow (az) perfect fluid [55] first-order dissipative fluid (az) J. D Bjorken, born 1934 156 Flows of relativistic fluids Appendices A PPENDIX A Basic elements of thermodynamics To be written! U = T S − PV + µN (A.1) dU = T dS − P dV + µ dN (A.2) e + P = T s

+ µn (A.3) de = T ds + µ dn . (A.4) dP = s dT + n dµ (A.5) Die letztere Gleichung folgt aus   1 U U de = d = dU − 2 dV V V V     T P µ TS P µN S N = dS − dV + dN − 2 dV + dV − 2 dV = T d + µd , V V V V V V V V wobei die Relation dU = T dS − P dV + µ dN benutzt wurde.  A PPENDIX B Tensors on a vector space In this Appendix, we gather mathematical definitions and results pertaining to tensors. The purpose is mostly to introduce the “modern”, geometrical view on tensors, which defines them by their action on vectors or one-forms, i.e in a coordinate-independent way (Sec B1), in contrast to the “old” definition based on their behavior under basis transformations (Sec. B2) The reader is assumed to already possess enough knowledge on linear algebra to know what are vectors, linear (in)dependence, (multi)linearity, matrices. Similarly, the notions of group, field, application/function/mapping. are used without further mention In the remainder of

these lecture notes, we actually consider tensors on real vector spaces, i.e for which the underlying base field K of scalars is the set R of real numbers; here we remain more general. Einstein’s summation convention is used throughout B.1 Vectors, one-forms and tensors B.11 Vectors . are by definition the elements ~c of a vector space V , ie of a set with 1) a binary operation (“addition”) with which it is an Abelian group, and 2) a multiplication with “scalars”elements of a base field Kwhich is associative, has an identity element, and is distributive with respect to both additions on V and on K. Introducing a basis B = {~ei }, i.e a family of linearly independent vectors that span the whole space V , one associates to each vector ~c its uniquely defined components {ci }, elements of the base field K, such that ~c = ci~ei . (B.1) If the number of vectors of a basis is finitein which case this holds for all bases and equal to some integer Dwhich is the same for all bases,

the space V is said to be finite-dimensional and D is its dimension (over K): D = dim V . We shall assume that this is the case in the remainder of this Section. B.12 One-forms . on a vector space V are the linear applications, hereafter denoted as h, from V into the e base field of scalars K. The set of 1-forms on V , equipped with the “natural” addition and scalar multiplication, is itself a vector space over the field K, denoted by V ∗ and said to be dual to V . If V is finite-dimensional, so is V ∗ , with dim V ∗ = dim V . Given a basis B = {~ei } in V , one can then construct its dual basis B ∗ = {j } in V ∗ such that e j (~ei ) = δij , (B.2) e where δij denotes the usual Kronecker delta symbol. 161 B.1 Vectors, one-forms and tensors The components of a 1-form h on a given basis will be denoted as {hj }: e h = hj j . e e (B.3) Remarks: ∗ The choice of notations, in particular the position of indices, is not innocent! Thus, if {j } e denotes the

dual base to {~ei }, the reader can trivially check that ci = i (~c) and hj = h(~ej ). (B.4) e e ∗ In the “old” language, the vectors of V resp. the 1-forms of V ∗ were designated as “contravariant vectors” resp. “covariant vectors” or “covectors”, and their coordinates as “contravariant” resp ”covariant” coordinates The latter two, applying to the components, remain useful short denominations, especially when applied to tensors (see below). Yet in truth they are not different components of a same mathematical quantity, but components of different objects between which a “natural” correspondence was introduced, in particular by using a metric tensor as in § B.14 B.13 Tensors B.13 a Definition and first results :::::::::::::::::::::::::::::::::: Let V be a vector space with base field K, and m, n denote two nonnegative integers. The multilinear applications of m one-formselements of V ∗ and n vectorselements of V into
m K are referred to as the

tensors of type n on V , where linearity should hold with respect to every argument. The integer m + n is the order (or often, but improperly, rank ) of the tensor Already known objects arise as special cases of this definition when either m or n is zero:
• the 00 -tensors are simply the scalars of the base field K;
• the 10 -tensors coincide with vectors;(59)

• the 01 -tensors are the one-forms. More generally, the n0 -tensors are also known as (multilinear) n-forms
• Eventually, 20 -tensors are sometimes called “bivectors” or “dyadics”. Tensors will generically be denoted as T , irrespective of their rank, unless the latter is 0 or 1. A tensor may be symmetric or antisymmetric under the exchange of two of its arguments, either both vectors or both 1-forms. Generalizing, it may be totally symmetricas eg the metric tensor we shall encounter below, or antisymmetric. An instance of the latter case is the determinant, which is the only (up to a multiplicative

factor) totally antisymmetric D-form on a vector space of dimension D.
∗ m ∗ n 0 0 Remark: Consider a m n -tensor T : (V ) ×(V ) K, and let m ≤ m, n ≤ n be two nonnegative integers. For every m0 -uplet of one-forms {hi } and n0 -uplet of vectors {~cj }and corresponding e we take for simplicity the first onesthe object multiplets of argument positions, although here
T h1 , . , hm0 , · , , · ; ~c1 , , ~cn0 , · , , · , e e where the dots denote “empty” arguments, can be applied to m − m0 one-forms and n − n0 vectors 0 0 to yield a scalar. That is, the tensor T induces a multilinear application(60) from (V ∗ )m × (V ∗ )n
0 into the set of m−m -tensors. n−n
0 1 For example, the 1 -tensors are in natural correspondence with the linear applications from V into V , i.e in turn with the square matrices of order dim V (59) (60) More accurately, they are the elements of the double dual of V , which is always homomorphic to V . Rather, the number

of such applications is the number of independentunder consideration of possible symmetriescombinations of m0 resp. n0 one-form resp vector arguments 162 Tensors on a vector space B.13 b Operations on tensors :::::::::::::::::::::::::::::: The tensors of a given type, with the addition and scalar multiplication inherited from V , form a vector space on K. Besides these natural addition and multiplication, one defines two further operations on tensors, the outer product or tensor productwhich increases the rankand the contraction, which decreases the rank.

0 m0 Consider two tensors T and T0 , of respective types m n and n0 . Their outer product T ⊗ T is
0 satisfying for every (m + m0 )-uplet (h1 , . , hm , , hm+m0 ) of 1-forms and a tensor of type m+m n+n0 e e e every (n + n0 )-uplet (~c1 , . , ~cn , , ~cn+n0 ) of vectors the identity
T ⊗ T0 h1 , . , hm+m0 ; ~c1 , , ~cn+n0 =

e e T h1 , . , hm ; ~c1 , , ~cn T0 hm+1 , , hm+m0 ; ~cn+1 , ,

~cn+n0 e e e e For instance, the outer product of two 1-forms h, h0 is a 2-form h ⊗ h0 such that for every pair e e of two vectors ~c, ~c 0 is a e e the outer product of
vectors (~c, ~c 0 ), h ⊗ h0 (~c, ~c 0 ) = h(~c) h0 (~c 0 ). In turn, e e e e 2 -tensor ~ c ⊗ ~c0 such that , h0 ), ~c ⊗ ~c0 (h, h 0 ) = h(~c) h0 (~c 0 ). 0
for every pair of 1-forms (h e and e n one-forms are e e m ofe me vectors Tensors of type n that can be written as outer products sometimes called simple tensors.
Let T be a m n -tensor, where both m and n are non-zero. To define the contraction over its j-th one-form and k-th vector arguments, the easiestapart from introducing the tensor componentsis to write T as a sum of simple tensors. By applying in each of the summand the k-th one-form to
m−1 the j-th vector, which gives a number, one obtains a sum of simple tensors of type n−1 , which is the result of the contraction operation. Examples of contractions will be given after the metric

tensor has been introduced. B.13 c Tensor coordinates ::::::::::::::::::::::::::: Let {~ei } resp. {j } denote bases on a vector space V of dimension D resp on its dual V ∗ in principle, they needenot be dual to each other, although using dual bases is what is implicitly always done in practiceand m, n be two nonnegative integers. The Dm+n simple tensors {~ei1 ⊗ · · · ⊗
~eim ⊗ j1 ⊗ · · · ⊗ jn }, where each ik or jk runs from 1 to D, e e form a basis of the tensors of type m n . The components of a tensor T on this basis will be denoted Ti1 .im j1 jn }: as {T T = T i1 .im j1 jn ~ei1 ⊗ · · · ⊗ ~eim ⊗ j1 ⊗ · · · ⊗ jn , (B.5a) e e where T i1 .im j1 jn = T (i1 , , im ;~ej1 , ,~ejn ) (B.5b) e e The possible symmetry or antisymmetry of a tensor with respect to the exchange of two of its arguments translates into the corresponding symmetry or antisymmetry of the components when exchanging the respective indices. In turn, the contraction of T over

its j-th one-form and k-th vector arguments yields the tensor with components T .ij−1 ,`,ij+1 , jk−1 ,`,jk+1 , , with summation over the repeated index ` B.14 Metric tensor Nondegenerate(61) symmetric bilinear forms play an important role, as they allow one to introduce a further structure on the vector space V , namely an inner product.(62) Accordingly, let {j } denote a basis on the dual space V ∗ . A 2-form g = gij i ⊗ j is a metric e e e tensor on V if it is symmetrici.e g(~a, ~b) = g(~b, ~a) for all vectors ~a, ~b, or equivalently gij = gji (61) This will be introduced 4 lines further down as a condition on the matrix with elements gij , which is equivalent to stating that for every non-vanishing vector ~a there exists ~b such that g(~a, ~b) 6= 0. (62) More precisely, an inner product if g is (positive or negative) definite, a semi-inner product otherwise. 163 B.1 Vectors, one-forms and tensors for all i, jand if the square matrix with elements gij is regular.

The number g(~a, ~b) is then also denoted ~a · ~b, which in particularly gives
(B.6) gij = g ~ei ,~ej = ~ei · ~ej , where {~ei } is the basis dual to {j }. e Since the D × D-matrix with elements gij is regular, it is invertible. Let g
ij denote the elements of its inverse matrix: gij g jk = δik , g ij gjk = δki . The D2 scalars g ij define a 20 -tensor g ij ~ei ⊗~ej , the inverse metric tensor , denoted as g−1 . Using results on symmetric matrices, the square matrix
with elements gij is diagonalizablei.e one can find an appropriate basis {~ei } such that g ~ei ,~ej = 0 for i 6= j. Since g is nondegenerate, the eigenvalues are non-zero: at the cost of multiplying the basis vectors {~ei } by a numerical factor,
one may demand that every g ~ei ,~ei be either +1 or −1, which yields the canonical form gij = diag(−1, . , −1, 1, , 1) (B.7) for the matrix representation of the components of the metric tensor. In that specific basis, the component g ij of g−1

coincides with gij , yet this does not hold in an arbitrary basis. Role of g in tensor algebra :::::::::::::::::::::::::::: In agreement with the remark at the end of § B.13 a, for any given vector ~c = ci~ei the object g(~c, ) maps vectors into the base field K, i.e it is a one-form c = cj j , such that e e cj = c(~ej ) = g(~c,~ej ) = g(ci~ei ,~ej ) = ci gij . (B.8a) e That is, a metric tensor g provides a mapping from vectors onto one-forms. Reciprocally its inverse metric tensor g−1 maps one-forms onto tensors, leading to the relation ci = g ij cj . (B.8b) Generalizing, a metric tensor and its inverse thus allow one “to lower

or to raise indices”, which m∓1 on a tensor of type are operations mapping a tensor of type m n n±1 , respectively. Remarks: ∗ Lowering resp. raising an index actually amounts to an outer product with g resp g−1 followed by the contraction of two indices. For instance ~c = ci~ei outer product contraction ~c ⊗ g = ci gjk ~ei ⊗ j

⊗ k 7− e e where the first and second arguments of ~c ⊗ g have been contracted. 7− c = ci gik k = ck k e e e ∗ Generalizing the “dot product” notation for the inner product defined by the metric tensor, the contraction is often also denoted with a dot product. For example, for a 2-form T and a vector ~c

T · ~c = T ij i ⊗ j · ck~ek = T ij cj i , e e e where we implicitly used Eq. (B2) Note that for the dot-notation to be unambiguous, it is better if T is symmetric, so that which of its indices is being contracted plays no role. Similarly, if T denotes a dyadic tensor and T0 a 2-form

T · T0 = T ij ~ei ⊗ ~ej · T0kl k ⊗ l = T ij T0jl ~ei ⊗ l , e e e which is different from T0 · T if the tensors are not symmetric. The reader may even find in the literature the notation T : T0 ≡ T ij T0ji , involving two successive contractions. 164 Tensors on a vector space B.2 Change of basis 0 Let B = {~ei } and B 0 = {~ej 0 } denote two bases of the

vector space V , and B ∗ = {i }, B 0∗ = {j } e the corresponding dual bases on V ∗ . The basis vector of B 0 can be expressed in terms of thosee of B with the help of a non-singular matrix Λ with elements Λi j 0 such that ~ej 0 = Λi j 0~ei . (B.9) Remark: Λ is not a tensor, for the two indices of its elements refer to two different baseswhich is
emphasized by the use of one primed and one unprimed indexwhile both components of a 1 -tensor are with respect to the “same” basis.(63) 1 0 Let Λk i denote the elements of the inverse matrix Λ−1 , that is 0 0 0 and Λi k0 Λk j = δji . Λk i Λi j 0 = δjk0 0 One then easily checks that the numbers Λk i govern the change of basis from B ∗ to B 0∗ , namely 0 0 j = Λj i i . e e Accordingly, each “vector” component transforms with Λ−1 : 0 0 cj = Λj i ci , 0 0 0 (B.10) 0 T j1 .jm = Λj1i1 · · · Λjmim T i1 im (B.11) In turn, every “1-form” component transforms with Λ: hj 0 = Λi j

0 hi , T j10 .jn0 = Λi1j10 · · · Λinjn0 T i1 in (B.12) One can thus obtain the coordinates of an arbitrary tensor in any basis by knowing just the transformation of basis vectors and one-forms. Bibliography for Appendix B • Your favorite linear algebra textbook. • A concise reminder can e.g be found in Nakahara [56], Chapter 22 • A more extensiveand elementarytreatment, biased towards geometrical applications of linear algebra, is provided in Postnikov [57]:(64) see e.g Lectures 1 (beginning), 4–6 & 18 (63) (64) Or rather, with respect to a basis and its dual. The reader should be aware that some of the mathematical terms usedas translated from the Russianare non-standard, e.g (linear, bilinear) “functional” for form or “conjugate” (space, basis) for dual A PPENDIX C Tensor calculus Continuum mechanics, and in particular fluid dynamics, is a theory of (classical) fields. The latter may be scalars, vectors or more generally tensorsmainly of degree at

most 2, whose dynamical behavior is governed by partial differential equations, which obviously involve various derivatives of tensorial quantities. When describing vector or tensor fields by their respective components on appropriate (local) bases, the basis vectors or tensors may actually vary from point to point. Accordingly, care must be taken when differentiating with respect to the space coordinates: instead of the usual partial derivatives, the quantities that behave in the expected manner are rather covariant derivatives (Sec. C1), which are the main topic of this Appendix To provide the reader with some elementary background on the proper mathematical framework to discuss vector and tensor fields and their differentiation, some basic ideas of differential geometry are gathered in Sec. C2, C.1 Covariant differentiation of tensor fields The purpose of this Section is to introduce the covariant derivative, which is the appropriate mathematical quantity measuring the spatial rate

of change of a field on a space, irrespective of the choice of coordinates on that space. The notion is first introduced for vector fields (Sec C11) and illustrated on the example of vector fields on a plane (Sec. C12) The covariant derivative of tensors of arbitrary type, in particular of one-forms, is then given in Sec. C13 Eventually, the usual differential operators of vector analysis are discussed in Sec. C14 Throughout this Section, we mostly list recipes, without providing proofs or the given results, nor specifying for example in which space the vector or tensor fields “live”. These more formal issues will be shortly introduced in Sec. C2 C.11 Covariant differentiation of vector fields Consider a set M of points generically denoted by P , possessing the necessary properties so that the following features are realized: (a) In a neighborhood of every point P ∈ M, one can find a system of local coordinates {xi (P )}. (b) It is possible to define functions on M with

sufficient smoothness properties, as e.g differentiable functions (c) At each point P ∈ M, one can attach vectorsand more generally tensors. Let {~ei (P )} denote a basis of the vectors at P . From the physicist’s point of view, the above requirements mean that we want to be able to define scalar, vector or tensor fields at each point [property (c)], that depend smoothly on the position [property (b)], where the latter can be labeled by local coordinates [property (a)]. Mathematically, it will be seen in Sec. C2 that the proper framework is to look at a differentiable manifold and its tangent bundle. 166 Tensor calculus Before we go any further, let us emphasize that the results we state hereafter are independent of the dimension n of the vectors, from 1 to which the indices i, j, k, l. run In addition, we use Einstein’s summation convention throughout. Assuming the above requirements are fulfilled, which we now do without further comment, we in addition assume that the

local basis {~ei (P )} at every point is that which is “naturally induced” by the coordinates {xi (P )},(65) and that for every possible i the mapping P 7 ~ei (P ) defines a continuous, and even differentiable vector field on M.(66) The derivative of ~ei at P with respect to any of the (local) coordinate direction xk is then itself a vector “at P ”, which may thus be expanded on the basis {~el (P )}: denoting by Γlik (P ) its coordinates ∂~ei (P ) (C.1) = Γlik (P )~el (P ). ∂xk  The numbers Γlik , which are also alternatively denoted as i lk , are called Christoffel symbols (of the second type) or connection coefficients. Remark: The reader should remember that the local coordinates also depend on P , i.e a better notation for the left hand side of Eq. (C1)and for every similar derivative in the followingcould be ∂~ei (P )/∂xk (P ). Let now ~c(P ) be a differentiable vector field defined on M, whose local coordinates at each point will be denoted by ci (P ) [cf. Eq

(B1)]: ~c(P ) = ci (P )~ei (P ). (C.2) The spatial rate of change in ~c between a point P and a neighboring point P 0 situated in the xk -direction with respect to P is given by dci (P ) ∂~c(P ) = ~ei (P ), ∂xk dxk where the component along ~ei (P ) is the so-called covariant derivative ∂ci (P ) dci (P ) = + Γilk (P )cl (P ). dxk ∂xk (C.3a) (C.3b) Remark: The covariant derivative dci /dxk is often denoted by ci;k , with a semicolon in front of the index (or indices) related to the direction(s) along which one differentiates. In contrast, the partial derivative ∂ci /∂xk is then written as ci,k , with a comma. That is, Eq (C3b) is recast as ci;k (P ) = ci,k (P ) + Γilk (P )cl (P ). (C.3c) The proof of Eqs. (C3) is rather straightforward Differentiating relation (C2) with the product rule first gives ∂~c(P ) ∂ci (P ) ei (P ) ∂ci (P ) i ∂~ = ~ e (P ) + c = ~ei (P ) + ci (P ) Γlik (P )~el (P ) i ∂xk ∂xk ∂xk ∂xk where we have used the derivative (C.1)

In the rightmost term, the dummy indices i and l may be relabeled as l and i, respectively, yielding ci Γlik ~el = cl Γilk ~ei , i.e ∂~c(P ) dci (P ) ∂ci (P ) l i (P )~ e (P ) = = ~ e (P ) + c (P ) Γ ~ei (P ). i i lk ∂xk ∂xk dxk
One can show that the covariant derivatives dci (P )/dxk are the components of a 11 -tensor field, the (1-form-)gradient of the vector field ~c, which may be denoted by ∇~c. On the other hand, neither e the partial derivative on the right hand side of Eq. (C3b) nor the Christoffel symbols are tensors. (65) (66) This requirement will be made more precise in Sec. C2 This implicitly relies on the fact that the vectors attached to every point P ∈ M all have the same dimension. 167 C.1 Covariant differentiation of tensor fields The Christoffel symbols can be expressed in terms of the (local) metric tensor g(P ), whose components are in agreement with relation (B.6) given by(67) gij (P ) = ~ei (P ) · ~ej (P ), (C.4) and of its partial

derivatives. Thus Γilk (P )   ∂gpl (P ) ∂gpk (P ) ∂gkl (P ) 1 ip = g (P ) + − 2 ∂xp ∂xk ∂xl (C.5) with g ip (P ) the components of the inverse metric tensor g−1 (P ). This relation shows that Γilk (P ) is symmetric under the exchange of the lower indices l and l, i.e Γikl (P ) = Γilk (P ) C.12 Examples: differentiation in Cartesian and in polar coordinates To illustrate the results introduced in the previous Section, we calculate the derivatives of vector fields defined at each point of the real plane R2 , which plays the role of the set M. C.12 a Cartesian coordinates As a first, trivial example, let us associate to each point P ∈ R2 local coordinates x1 (P ) = x, 2 x (P ) = y that coincide with the usual global Cartesian coordinates on the plane. Let ~e1 (P ) = ~ex , ~e2 (P ) = ~ey denote the corresponding local basis vectorswhich actually happen to be the same at every point P , i.e which represent constant vector fields Either by writing down the

vanishing derivatives ∂~ei (P )/∂xk , i.e using Eq (C1), or by invoking relation (C.5)where the metric tensor is trivial: g11 = g22 = 1, g12 = g21 = 0 everywhere, one finds that every Christoffel symbol vanishes. This means [Eq (C3b)] that covariant and partial derivative coincide. which is why one need not worry about “covariant differentiation” when working in Cartesian coordinates. :::::::::::::::::::::::::::::: C.12 b Polar coordinates :::::::::::::::::::::::::: It is thus more instructive to associate to each point P ∈ R2 , with the exception of the origin, 0 0 polar coordinates x1 = r ≡ xr , x2 = θ ≡ xθ . The corresponding local basis vectors are ( ~er (r, θ) = cos θ~ex + sin θ~ey (C.6) ~eθ (r, θ) = −r sin θ~ex + r cos θ~ey . To recover the usual inner product on R2 , the metric tensor g(P ) should have components grr (r, θ) = 1, gθθ (r, θ) = r2 , grθ (r, θ) = gθr (r, θ) = 0. (C.7a) g rθ (r, θ) = g θr (r, θ) = 0. (C.7b) That is, the

components of g−1 (P ) are g rr (r, θ) = 1, g θθ (r, θ) = 1 , r2 Computing the derivatives 1 ∂~eθ (r, θ) ∂~eθ (r, θ) ∂~er (r, θ) ~ ∂~er (r, θ) 1 = ~eθ (r, θ), = −r~er (r, θ) = 0, = ~eθ (r, θ), r r θ ∂x r ∂x r ∂x ∂xθ and using Eq. (C1), or relying on relation (C5), one finds the Christoffel symbols 1 Γθrθ = Γθθr = , Γrθθ = −r, Γrrθ = Γrθr = 0, Γθθθ = 0 r where for the sake of brevity the (r, θ)-dependence of the Christoffel symbols was dropped. Γrrr = Γθrr = 0, (67) Remember that the metric tensor g actually defines the inner product. (C.8) 168 Tensor calculus Remarks: ∗ The metric tensor in polar coordinates (C.7a) has signature (0, 2)ie 0 negative and 2 positive eigenvalues, just like it has in Cartesian coordinates: the signature of the metric (tensor) is independent of the choice of coordinates if it defines the same inner product. ∗ It is also interesting to note that the Christoffel symbols for polar

coordinates (C.8) are not all zero, while this is the case for the Christoffel symbols in Cartesian coordinates. This shows that the Christoffel symbols are not the components of a tensora tensor which is identically zero in a basis remains zero in any basis. Consider now a constant vector field ~c(P ) = ~c(r, θ) = ~ex . Obviously, it is unchanged when going from any point (r, θ) to any neighboring point, i.e a meaningful derivative along either the r or θ direction should identically vanish. Let us write sin θ ~c(r, θ) = ~ex = cos θ~er (r, θ) − ~eθ (r, θ) = cr (r, θ)~er (r, θ) + cθ (r, θ)~eθ (r, θ). r The partial derivatives ∂cr /∂xθ , ∂cθ /∂xr and ∂cθ /∂xθ are clearly non-vanishing. On the other hand, all covariant derivatives are identically zero: omitting the variables, one finds ∂cr dcθ ∂cθ sin θ 1 (− sin θ) dcr = = 0, = + Γθθr cθ = 2 + = 0, r r r r dx ∂x dx ∂x r r r i.e d~c/dxr = ~0, and dcθ (− sin θ) cos θ 1 ∂cr ∂cθ

dcr = 0, + cos θ = 0, = + Γrθθ cθ = − sin θ − r = + Γθrθ cr = − θ θ θ r r r dx ∂x dx ∂xθ θ ~ i.e d~c/dx = 0 Thus the covariant derivatives give the expected result, while the partial derivatives with respect to the coordinates do not. C.13 Covariant differentiation of general tensor fields C.13 a Scalar fields :::::::::::::::::::: scalar field f (P ) ∂f (P ) df (P ) = . k dx ∂xk (C.9) ∂hj (P ) dhj (P ) = − Γljk (P )hl (P ). dxk ∂xk (C.10) C.13 b One-forms :::::::::::::::::: one-form field h(P ) = hj (P ) j (P ) e e C.13 c
:::::::::::::::::::::::::: Tensors of arbitrary type ::::::: m -tensor field T (P ) n m Tij11···i dT .jn (P ) dxk = m Tij11···i ∂T .jn (P ) ∂xk i ···i m Tilj12···i − Γlj1 k (P )T .jn (P ) C.14 Gradient, divergence, Laplacian to be completed! l im 2 ···im Tj11 .jm−1 Tli (P ) + Γikl1 (P )T j1 .jn (P ) + · · · + Γkl (P )T n − ··· − m Tij11···i Γljn k (P )T .jn−1 l (P )

(C.11) C.2 Beginning of elements of an introduction to differential geometry 169 C.2 Beginning of elements of an introduction to differential geometry attempt Bibliography for Appendix C • Nakahara [56], Chapter 5.1–53 & 71–72 • Postnikov [57]:(64) see e.g Lectures 1 (beginning), 4–6 & 18 (C.12) A PPENDIX D Elements on holomorphic functions of a complex variable D.1 Holomorphic functions D.11 Definitions A function Z = f (z) is defined to be complex-differentiable at a point z0 in its domain of definition if the limit f (z) − f (z0 ) (D.1) f 0 (z0 ) ≡ lim zz0 z − z0 exists independently of the direction along which z approaches z0 . If f is complex-differentiable at every point of an open set U resp. of a neighborhood of a point z0 , it is said to be holomorphic on U resp. at z0 D.12 Some properties D.12 a Cauchy–Riemann equations :::::::::::::::::::::::::::::::::::: Let P (x, y) resp. Q(x, y) denote the real resp imaginary part of a function f

(z = x +iy) of a complex variable: f (x+iy) = P (x, y) + iQ(x, y). (D.2) Theorem: f is holomorphic if and only if the Cauchy–Riemann equations ∂Q(x, y) ∂P (x, y) = ∂x ∂y and ∂P (x, y) ∂Q(x, y) =− ∂y ∂x (D.3) relating the first partial derivatives of its real and imaginary parts are satisfied. Equivalently, the relations (D.3) can be recast as df =0 dz̄ (D.4) where z̄ = x − iy. Corollary: A function f (z = x+iy) is holomorphic on a domain if and only if its real and imaginary parts are conjugate harmonic functions, i.e they obey the Cauchy–Riemann equations (D3) and the Laplace equations 4P (x, y) = 0 , 4Q(x, y) = 0 (D.5) on the domain. D.12 b Integration of holomorphic functions ::::::::::::::::::::::::::::::::::::::::::::: Z Z f (z) dz = C a b
f γ(t) γ 0 (t) dt (D.6) 171 D.2 Multivalued functions Cauchy’s integral theorem I f (z) dz = 0. (D.7) C Cauchy’s integral formula I 1 2πi f (z0 ) = f (z) dz. C z − z0 (D.8) D.2

Multivalued functions D.3 Series expansions D.31 Taylor series f (z) = ∞ X f (n)(z0 ) n! n=0 f (n)(z0 ) = n! 2πi (z − z0 )n (D.9) I f (z) dz, (z − z0 )n+1 C (D.10) which generalizes the Cauchy integral formula (D.8) to the successive derivatives of f zeroes D.32 Isolated singularities and Laurent series D.32 a Definitions :::::::::::::::::: isolated singularity removable singularity pole of order m essential singularity D.32 b Laurent series :::::::::::::::::::::: ∞ X f (z) = an (z − z0 )n (D.11) n=−∞ with 1 an = 2πi where C denotes a−1 residue D.33 Singular points I f (z) dz n+1 C (z − z0 ) (D.12) 172 Elements on holomorphic functions of a complex variable D.4 Conformal maps Function Z = f (z) defines mapping from plane of complex variables z = x + iy to plane of complex Z. Such a function is said to be a conformal map if it preserves angles locally If a function Z = f (z) is holomorphic at z0 and such that f 0 (z0 ) 6= 0, it is

invertible in a neighborhood of z0 , and f and its inverse F define a conformal mapping between the planes z and Z. 0 Proof: dZ = |f 0 (z0 )| ei arg f (z0 ) dz. Singular point: f 0 (z0 ) = 0: if zero of n, angles are multiplied by n + 1 in transformation z Z. φ(z) complex potential on z-plane. Then Φ(Z) ≡ φ(F (Z)) potential on Z-plane, with velocity w(F (Z))F 0 (Z) Bibliography for Appendix D • Cartan [58], Chapters II, III & VI; • Whittaker & Watson [59], Chapters 4.6, 51–52 & 56–57 Bibliography [1] T. E Faber, Fluid dynamics for physicists (University Press, Cambridge, 1995) [2] E. Guyon, J-P Hulin, L Petit, C D Mitescu, Physical hydrodynamics, 2nd ed (University Press, Oxford, 2015). [3] L. Landau, E Lifshitz, Course of theoretical physics Vol VI : Fluid mechanics, 2nd ed (Pergamon, Oxford, 1987) [4] L. Landau, E Lifschitz, Lehrbuch der theoretischen Physik Band VI : Hydrodynamik , 5 ed (Harri Deutsch, Frankfurt am Main, 1991). [5] A. Sommerfeld,

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